Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas
aa r X i v : . [ m a t h . P R ] M a y Probability Surveys
Vol. 4 (2007) 80–145ISSN: 1549-5787DOI:
Brownian excursion area, Wright’sconstants in graph enumeration, andother Brownian areas ∗ Svante Janson
Department of Mathematics, Uppsala UniversityPO Box 480SE-751 06 Uppsala, Swedene-mail: [email protected] url: ∼ svante/ Abstract:
This survey is a collection of various results and formulas bydifferent authors on the areas (integrals) of five related processes, viz. Brow-nian motion, bridge, excursion, meander and double meander; for the Brow-nian motion and bridge, which take both positive and negative values, weconsider both the integral of the absolute value and the integral of thepositive (or negative) part. This gives us seven related positive randomvariables, for which we study, in particular, formulas for moments andLaplace transforms; we also give (in many cases) series representations andasymptotics for density functions and distribution functions. We furtherstudy Wright’s constants arising in the asymptotic enumeration of con-nected graphs; these are known to be closely connected to the moments ofthe Brownian excursion area.The main purpose is to compare the results for these seven Brownianareas by stating the results in parallel forms; thus emphasizing both thesimilarities and the differences. A recurring theme is the Airy function whichappears in slightly different ways in formulas for all seven random variables.We further want to give explicit relations between the many different similarnotations and definitions that have been used by various authors. Thereare also some new results, mainly to fill in gaps left in the literature. Someshort proofs are given, but most proofs are omitted and the reader is insteadreferred to the original sources.
AMS 2000 subject classifications:
Primary 60J65, ; secondary 05C30,60G15.Received April 2007.
Contents ∗ This is an original survey paper 80 . Janson/Brownian areas and Wright’s constants B ex . . . . . . . . . . . . . . . . . . . . . . . . . 10220 Brownian bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10221 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10722 Brownian meander . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11023 Brownian double meander . . . . . . . . . . . . . . . . . . . . . . . . . 11324 Positive part of a Brownian bridge . . . . . . . . . . . . . . . . . . . . 11625 Positive part of a Brownian motion . . . . . . . . . . . . . . . . . . . . 12026 Convolutions and generating functions . . . . . . . . . . . . . . . . . . 12427 Randomly stopped Brownian motion . . . . . . . . . . . . . . . . . . . 12428 A comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12729 Negative moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127A The integrated Airy function . . . . . . . . . . . . . . . . . . . . . . . 132B The Mellin transform of a Laplace transform . . . . . . . . . . . . . . 133C Feynman-Kac formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 135C.1 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 137C.2 Brownian bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . 138C.3 Brownian meander . . . . . . . . . . . . . . . . . . . . . . . . . . 140C.4 Brownian excursion . . . . . . . . . . . . . . . . . . . . . . . . . . 141C.5 Brownian double meander . . . . . . . . . . . . . . . . . . . . . . 142Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
1. Introduction
This survey started as an attempt to better understand two different results,viz. the asymptotic enumeration of connected graphs with a given number ofindependent cycles by Wright [58; 59] and the formulas for moments of theBrownian excursion area by Louchard [34], Tak´acs [44] and others, and thesurprising connection between these two seemingly unrelated results found (andexplained) much later by Spencer [43], who showed that the same sequence ofconstants appears in both results (see (36) below). The literature may seemconfusing (to me at least), however, because different authors, and sometimesthe same author in different papers, have used not only different notations for . Janson/Brownian areas and Wright’s constants the same constants but also several different related constants. I therefore setout to collect the various definitions and notations, and to list the relationsbetween them explicitly. I was further inspired by the paper by Flajolet andLouchard [17], studying further related properties of the distribution.After doing this, I realized that many of the results for Brownian excur-sion area have close parallels for the integrals of (the absolute value of) otherrelated processes, viz. Brownian bridge, Brownian motion, Brownian meanderand Brownian double meander, and also for the positive parts of Brownianmotion and Brownian bridge (see Sections 20–25). In particular, there are sim-ilar recursion formulas for moments and similar expressions for double Laplacetransforms involving the Airy function, but the details differ between the sevendifferent Brownian areas. I therefore decided to collect various results for theseseven different areas of Brownian processes together so that both the similari-ties and the differences would become clear. One important source of results andinspiration for this is the series of papers by Tak´acs [44; 45; 46; 47; 48; 49; 50];another, treating different cases in a unified way, is Perman and Wellner [38]. Inthe course of doing this, I have also added some minor new results motivated byparallel results. For example, much of Section 29 and the related Appendix B,which both generalize results by Flajolet and Louchard [17], seem to be new, asdo many of the explicit asymptotic results as x → . Janson/Brownian areas and Wright’s constants The Brownian excursion area is introduced in Section 2 and some equivalentdescriptions are given in Section 3. The results on graph enumeration are given inSection 4, the connection with the Brownian excursion area is given in Section 5,and various aspects of these results and other related results on the Brownianexcursion area are discussed in Sections 6–18. Integrals of higher powers ofa Brownian excursion are discussed briefly in Section 19. The other types ofBrownian processes are studied in Sections 20–25; these sections are intendedto be parallel to each other as much as possible. Furthermore, in Sections 24and 25, results are given both for the positive parts of a Brownian bridge andmotion, respectively, and for the joint distribution of the positive and negativeparts. Sections 26 and 27 give more information on the relation between theseven different Brownian areas (the reader may prefer to read these sectionsbefore reading about the individual variables), and Section 28 compares themnumerically. Finally, Section 29 discusses negative moments of the Brownianareas.The appendices contain some general results used in the main part of thetext.The first few moments of the seven different Brownian areas studied hereare given explicitly in 11 tables throughout the paper together with the firstelements in various related sequences of constants. Most of these values, andsome other expressions in this paper, have been calculated using
Maple .We let B ( t ) be a standard Brownian motion on the [0 , ∞ ) with B (0) =0 (usually we consider only the interval [0 , B br ( t ) can be defined as B ( t ) on the interval [0 ,
1] conditionedon B (1) = 0; a (normalized) Brownian meander B me ( t ) as B ( t ) on the interval[0 ,
1] conditioned on B ( t ) ≥ t ∈ [0 , B ex ( t ) as B ( t ) on the interval [0 ,
1] conditioned on both B (1) = 0 and B ( t ) ≥ t ∈ [0 , T > T = 1), and g T := max { t < T : B ( t ) = 0 } and d T := min { t > T : B ( t ) = 0 } are the zeros of B nearest before and after T , thenconditioned on g T and d T , the restrictions of B to the intervals [0 , g T ], [ g T , T ]and [ g T , d T ] are, respectively, a Brownian bridge, a Brownian meander and aBrownian excursion on these intervals; the normalized processes on [0 ,
1] studiedin this paper can be obtained by standard Brownian rescaling as g − / T B ( tg T ),( T − g T ) − / B ( g T + t ( T − g T )) and ( d T − g T ) − / B ( g T + t ( d T − g T )). (Seealso Section 27, where we consider Brownian motion, bridges, meanders andexcursions on other intervals than [0 , B br ( t ) = B ( t ) − tB (1), t ∈ [0 , . Janson/Brownian areas and Wright’s constants While the Brownian motion, bridge, meander and excursion have been ex-tensively studied, the final process considered here, viz. the Brownian doublemeander, has not been studied much; Majumdar and Comtet [35] is the mainexception. In fact, the name has been invented for this paper since we havenot seen any given to it previously. We define the Brownian double meander by B dm ( t ) := B ( t ) − min ≤ u ≤ B ( u ); this is a non-negative continuous stochasticprocess on [0 ,
1] that a.s. is 0 at a unique point τ ∈ [0 , , τ ] and [ τ,
1] joined back to back(with the first one reversed), see Section 23.
2. Brownian excursion area and its moments
Let B ex denote a (normalized) Brownian excursion and B ex := Z B ex ( t ) d t, (1)the Brownian excursion area. Two variants of this are A := 2 / B ex (2)studied by Flajolet, Poblete and Viola [18] and Flajolet and Louchard [17] (theirdefinition is actually by the moments in (8) below), and ξ := 2 B ex (3)used in [21; 23; 22]. (Louchard [34] uses ξ for our B ex ; Tak´acs [44; 45; 46; 47; 48]uses ω + ; Perman and Wellner [38] use A excur ; Spencer [43] uses L . Flajolet andLouchard [17] use simply B for B ex , but since we in later sections will considerareas of other related processes too, we prefer to use B ex throughout the paperfor consistency.) Flajolet and Louchard [17] call the distribution of A the Airydistribution , because of the relations with the Airy function given later. (Butnote that the other Brownian areas in this paper also have similar connectionsto the Airy function.)This survey centres upon formulas for the moments of B ex and various ana-logues of them. The first formula for the moments of B ex was given by Louchard[34] (using formulas in [33]), who showed (using β n for E B n ex ) E B k ex = (36 √ − k √ π Γ((3 k − / γ k , k ≥ , (4)where γ k satisfies γ r = 12 r r − r + 1 / r + 1 / − r − X j =1 (cid:18) rj (cid:19) Γ(3 j + 1 / j + 1 / γ r − j , r ≥ . (5)Tak´acs [44; 45; 46; 47; 48] gives the formula (using M k for E B k ex ) E B k ex = 4 √ π − k/ k !Γ((3 k − / K k , k ≥ , (6) . Janson/Brownian areas and Wright’s constants where K = − / K k = 3 k − K k − + k − X j =1 K j K k − j , k ≥ . (7)(See also Nguyˆen Thˆe [36], where E B r ex is denoted M Dr and a Dr equals 2 / − r/ K r .)We will later, in Section 16, see that both the linear recurrence (4)–(5) andthe quadratic recurrence (6)–(7) follow from the same asymptotic series (98) in-volving Airy functions. We will in the sequel see several variations and analoguesfor other Brownian areas of both these recursions.Flajolet, Poblete and Viola [18] and Flajolet and Louchard [17] give theformula E A k = 2 √ π Γ((3 k − /
2) Ω k , k ≥ , (8)(and use it as a definition of the distribution of A ; they further use µ ( k ) and µ k , respectively, for E A k ). Here Ω k is defined by Flajolet and Louchard [17] byΩ := − k = (3 k − k Ω k − + k − X j =1 (cid:18) kj (cid:19) Ω j Ω k − j , k ≥ . (9)In particular, Ω = 1 / K k = Ω k k +1 k ! , k ≥ . (10)Flajolet, Poblete and Viola [18] define further ω k := Ω k k ! = 2 k +1 K k , k ≥ , (11) ω ∗ k := 2 k − ω k = 2 k − k ! Ω k = 2 k K k , k ≥ . (12)By (12), the recursion (9) translates to ω ∗ k = 2(3 k − ω ∗ k − + k − X j =1 ω ∗ j ω ∗ k − j , k ≥
1; (13)with ω ∗ := − / ω ∗ = 1. Note that ω ∗ k thus is an integer for k ≥
1. Thenumbers ω ∗ k are the same as ω ∗ k in Janson [21]. (This is a special case of ω ∗ kl in [21], but we will only need the case l = 0.) The sequence ( ω ∗ k ) is called theWright–Louchard–Tak´acs sequence in [18].By a special case of Janson [21, Theorem 3.3], E ξ k = 2 − k/ √ π k !Γ((3 k − / ω ∗ k , k ≥ , (14) . Janson/Brownian areas and Wright’s constants which is equivalent to (6) by (3) and (12).Further, (4) is equivalent to (8) and γ k = 18 k Ω k , k ≥ . (15)It is easily seen from (9) that 2 k Ω k is an integer for all k ≥
0, and thus by (15), γ k is an integer for k ≥
3. Other Brownian representations
Let B br denote a Brownian bridge. By Vervaat [54], B br ( · ) − min [0 , B br hasthe same distribution as a random translation of B ex , regarding [0 ,
1] as a circle.(That is, the translation by u is defined as B ex ( ⌊· − u ⌋ ).) In particular, theserandom functions have the same area: B ex := Z B ex ( t ) d t d = Z B br ( t ) d t − min ≤ t ≤ B br ( t ) . (16)This equivalence was noted by Tak´acs [44; 45], who considered the functionalof a Brownian bridge in (16) in connection with a problem in railway traffic.Darling [12] considered the maximum (denoted G by him) of the process Y ( t ) := B br ( t ) − Z B br ( s ) d s, ≤ t ≤ . (17)Since B br is symmetric, B br d = − B br , we obtain from (16) also B ex d = max ≤ t ≤ B br ( t ) − Z B br ( t ) d t = max ≤ t ≤ Y ( t ) . (18)Thus Darling’s G := max t Y ( t ) [12] equals B ex (in distribution). Y ( t ) is clearlya continuous Gaussian process with mean 0, and a straightforward calculationyields Cov (cid:0) Y ( s ) , Y ( t ) (cid:1) = ( | t − s | − ) − , s, t ∈ [0 , . (19)It follows that if we regard [0 ,
1] as a circle, or if we extend Y periodically to R , Y is a stationary Gaussian process, as observed by Watson [56]. Furthermore,by construction, the integral R Y ( t ) d t vanishes identically.
4. Graph enumeration
Let C ( n, q ) be the number of connected graphs with n given (labelled) verticesand q edges. Recall Cayley’s formula C ( n, n −
1) = n n − for every n ≥
1. Wright[58] proved that for any fixed k ≥ −
1, we have the analoguous asymptoticformula C ( n, n + k ) ∼ ρ k n n +(3 k − / as n → ∞ , (20) . Janson/Brownian areas and Wright’s constants for some constants ρ k given by ρ k = 2 (1 − k ) / π / Γ(3 k/ σ k , k ≥ − , (21)with other constants σ k given by σ − = − / σ = 1 / σ = 5 /
16, and thequadratic recursion relation σ k +1 = 3( k + 1)2 σ k + k − X j =1 σ j σ k − j , k ≥ . (22)Note the equivalent recursion formula σ k +1 = 3 k + 22 σ k + k X j =0 σ j σ k − j , k ≥ − . (23)Wright gives in the later paper [59] the same result in the form ρ k = 2 (1 − k ) / k π / ( k − k/ d k , k ≥ , (24)(although he now uses the notation f k = ρ k ; we have further corrected a typoin [59, Theorem 2]), where d = 5 /
36 and d k +1 = d k + k − X j =1 d j d k − j ( k + 1) (cid:0) kj (cid:1) , k ≥ . (25)See also Bender, Canfield and McKay [5, Corollaries 1 and 2], which gives theresult using the same d k and further numbers w k defined by w = π/ √ w k = (8 / / π ( k − k/ (cid:16) k e (cid:17) k/ d k , k ≥ , (26)so that ρ k = 3 / π / (cid:16) e k (cid:17) k/ w k . k ≥ . (27)(Wright [59] and Bender et al. [5] further consider extensions to the case k → ∞ ,which does not interest us here.)In the form ρ k = 2 − (5 k +1) / k +1 π / k !Γ(3 k/ d k , (28)(24) holds for all k ≥
0, with d = 1 / σ k = (cid:16) (cid:17) k +1 k ! d k , k ≥ . (29) . Janson/Brownian areas and Wright’s constants Next, define c k , k ≥
1, as in Janson, Knuth, Luczak and Pittel [24, § c k isthe coefficient for the leading term in an expansion of the generating functionfor connected graphs (or multigraphs) with n vertices and n + k edges. (Notethat c k = c k = ˆ c k in [24, § c k is denoted c k, − k in Wright [58] and b k inWright [59].) We have by Wright [59, § c k = (cid:16) (cid:17) k ( k − d k , k ≥ . (30)From [24, (8.12)] (which is equivalent to Wright [58, (7)]) follows the recursion3 rc r = (3 r − r − c r − + r − X j =0 j ( r − − j ) c j c r − − j , r ≥ , (31)where jc j is interpreted as 1 / j = 0. In other words, c = 5 /
24 and rc r = r (3 r − c r − + r − X j =1 j ( r − − j ) c j c r − − j , r ≥ . (32)By (29), (30) is equivalent to σ k = 32 kc k , k ≥ k = 0 with the interpretation 0 c = 1 / §§ k ≥ ρ k = 2 (1 − k ) / √ π Γ(3 k/ c k , k ≥ , (34)which clearly is equivalent to (21) and (24) by (33) and (30).Finally, we note that (20) can be written C ( n, n + k − ∼ ρ k − n n +3 k/ − k ≥ . (35)
5. The connection
The connection between Brownian excursion area and graph enumeration wasfound by Spencer [43], who gave a new proof of (35), and thus (20), that furthershows ρ k − = E B k ex k ! , k ≥ . (36)See also Aldous [2, § σ k − = 2 k K k , k ≥ , (37)and it is easily seen that (23) is equivalent to (7). Similarly, (9) is equivalent to(23) and Ω k = 2 k ! σ k − , k ≥ , (38)and (8) is equivalent to (21) by (2), (36) and (38). . Janson/Brownian areas and Wright’s constants
6. Further relations
Further relations are immediately obtained by combining the ones above; wegive some examples.By (38) and (12), or by comparing (13) and (23), ω ∗ k = 2 k σ k − , k ≥ , (39)and thus also, by (12), ω k = 2 σ k − , k ≥ . (40)By (38) and (33), or comparing (31) and (9), we findΩ k = 3( k − k ! c k − , k ≥ . (41)By (12) and (41), or by (39), (40) and (33), or by (31) and (13), ω ∗ k = 2 k − k − c k − , k ≥ , (42) ω k = 3( k − c k − , k ≥ . (43)By (29) and (37), d k − = (cid:16) (cid:17) k K k ( k − , k ≥ . (44)By (40) and (29), or by (11) and (44), ω k = 2 (cid:16) (cid:17) k ( k − d k − , k ≥ . (45)By (21) and (39), ρ k − = 2 − k/ √ π Γ((3 k − / σ k − = 2 − k/ √ π Γ((3 k − / ω ∗ k , k ≥ . (46)Note further that (34) can be written ρ k − = 2 − k/ √ π Γ(3( k − / c k − , k ≥ . (47)By (14) and (42), or by (3), (36) and (47), E ξ k = 2 − k/ √ π k !Γ((3 k − /
2) ( k − c k − = 2 − k/ √ π k !Γ(3( k − / c k − , k ≥ , (48)as claimed in Janson [22, Remark 2.5]. . Janson/Brownian areas and Wright’s constants
7. Asymptotics
Wright [59] proved that the limit lim k →∞ d k exists, and gave the approximation0 . / (2 π ),i.e. d k → π as k → ∞ . (49)See [24, p. 262] for further history and references.It follows by Stirling’s formula that for w k in (26), w k → k → ∞ . (50)Hence, by (27) ρ k ∼ / π / (cid:16) e k (cid:17) k/ as k → ∞ , (51)and, equivalently, ρ k − ∼ π − / k / (cid:16) e k (cid:17) k/ as k → ∞ . (52)By (36) follows further, as stated in Tak´acs [44; 45; 46; 47; 48], see also Jansonand Louchard [25], E B k ex ∼ √ k (cid:16) k e (cid:17) k/ as k → ∞ , (53)and equivalently, as stated by Flajolet and Louchard [17] and Chassaing andJanson [23], respectively, E A k ∼ / k (cid:16) k e (cid:17) k/ as k → ∞ , (54) E ξ k ∼ / k (cid:16) k e (cid:17) k/ as k → ∞ . (55)By (30) and (49), or by [24, Theorem 8.2], c k ∼ π (cid:16) (cid:17) k ( k − k → ∞ . (56)Further, by (29) and (49), σ k ∼ π (cid:16) (cid:17) k +1 k ! as k → ∞ , (57)and by (44) and (49), as stated in Tak´acs [44; 45; 46; 47; 48], K k ∼ π (cid:16) (cid:17) k ( k − k → ∞ . (58) . Janson/Brownian areas and Wright’s constants E B = 1 E B ex = √ π E B = 512 E B = 15 √ π E B = 2211008 E A = 1 E A = √ π E A = 103 E A = 15 √ π E A = 88463 γ = − γ = 9 γ = 405 γ = 65610 γ = 21749715 K = − K = 18 K = 564 K = 15128 K = 11054096Ω = − = 12 Ω = 54 Ω = 454 Ω = 331516 ω = − ω = 12 ω = 58 ω = 158 ω = 1105128 ω ∗ = − ω ∗ = 1 ω ∗ = 5 ω ∗ = 60 ω ∗ = 1105 ρ − = 1 ρ = √ π ρ = 524 ρ = 5 √ π ρ = 22124192 σ − = − σ = 14 σ = 516 σ = 1516 σ = 1105256 d = 16 d = 536 d = 536 d = 11057776 c = 524 c = 516 c = 11051152 Table 1
Some numerical values for the Brownian excursion and for graph enumeration.
Similarly by (57) and (39), (40) and (11), ω ∗ k ∼ π k ( k − k → ∞ , (59) ω k ∼ π (cid:16) (cid:17) k ( k − k → ∞ , (60)Ω k ∼ πk (cid:16) (cid:17) k ( k !) ∼ (cid:16) k e (cid:17) k as k → ∞ . (61)
8. Numerical values
Numerical values for small k are given in Table 1. See further Louchard [34]( E B k ex ), Tak´acs [44; 47; 48] ( E B k ex , K k ), Janson, Knuth, Luczak and Pittel [24,p. 259 or 262] ( c k ), Flajolet, Poblete and Viola [18, Table 1 and p. 503] ( E A k ,Ω k , ω k , ω ∗ k ), Flajolet and Louchard [17, Table 1] ( E A k , Ω k ), and Janson [21, p.343] ( ω ∗ k ). . Janson/Brownian areas and Wright’s constants
9. Power series
Define the formal power series C ( z ) := ∞ X r =1 c r z r . (62)By [24, § e C ( z ) = ∞ X r =0 e r z r , (63)where e r := (6 r )!2 r r (3 r )! (2 r )! = (cid:16) (cid:17) r Γ( r + 5 / r + 1 / πr ! = (cid:16) (cid:17) r (5 / r (1 / r r != 18 − r Γ(3 r + 1 / r + 1 / r ! . (64)(The last formula follows by the triplication formula for the Gamma function,or by induction.) We have e = 1, e = 5 / e = 385 / e = 85085 / e r are the coefficient for the leading term in an expansion of thegenerating function for all graphs (or multigraphs) with n vertices and n + k edges, see [24, §
7] (where e r is denoted e r ).
10. Linear recursions
From (63) follows the linear recursion, see [24, § c r = e r − r r − X j =1 jc j e r − j , r ≥ , (65)where e k are given explicitly by (64).By (41) together with (64) and simple calculations, (65) is equivalent to18 r Ω r = 12 r r − r + 1 / r + 1 / − r − X j =1 (cid:18) rj (cid:19) Γ(3 j + 1 / j + 1 /
2) 18 r − j Ω r − j , r ≥ , (66)given by Flajolet and Louchard [17], and, equivalently, see (15), γ r = 12 r r − r + 1 / r + 1 / − r − X j =1 (cid:18) rj (cid:19) Γ(3 j + 1 / j + 1 / γ r − j , r ≥ , (67)given by Louchard [34]; the latter was already given as (5) above. Changing theupper summation limit we can also write these as18 r Ω r = 6 r + 16 r − r + 1 / r + 1 / − r X j =1 (cid:18) rj (cid:19) Γ(3 j + 1 / j + 1 /
2) 18 r − j Ω r − j , r ≥ , (68) . Janson/Brownian areas and Wright’s constants γ r = 6 r + 16 r − r + 1 / r + 1 / − r X j =1 (cid:18) rj (cid:19) Γ(3 j + 1 / j + 1 / γ r − j , r ≥ . (69)By (10), these are further equivalent to the linear recursion in Tak´acs [44; 45] K r = 6 r + 12(6 r − α r − r X j =1 α j K r − j , r ≥ , (70)where, cf. (64), α j := 36 − j Γ(3 j + 1 / j + 1 / j ! = 2 − j e j , j ≥ . (71)(Perman and Wellner [38] and Majumdar and Comtet [35] use γ j for α j .)
11. Semi-invariants
Tak´acs [44] considers also the semi-invariants (cumulants) Λ n of B ex , given re-cursively from the moments M k := E B k ex by the general formulaΛ n = M n − n − X k =1 (cid:18) n − k (cid:19) M k Λ n − k , n ≥ . (72)For example, Λ = E B ex = p π/
8, Λ = Var( B ex ) = (10 − π ) /
24 and Λ = E B − E B E B ex + 2( E B ex ) = (8 π − √ π/
12. The Airy function
There are many connections between the distribution of B ex and the transcen-dental Airy function ; these connections have been noted in different contexts byseveral authors and, in particular, studied in detail by Flajolet and Louchard[17]. We describe many of them in the following sections.The Airy function Ai( x ) is defined by, for example, the conditionally conver-gent integral Ai( x ) := 1 π Z ∞ cos (cid:0) t / xt (cid:1) d t, −∞ < x < ∞ , (73)see e.g. Abramowitz and Stegun [1, § § x ) = 1 π (cid:16) x (cid:17) / K / (cid:16) x / (cid:17) , x > , (74)Ai( x ) = x / (cid:16) I − / (cid:16) x / (cid:17) − I / (cid:16) x / (cid:17)(cid:17) , x > , (75) . Janson/Brownian areas and Wright’s constants Ai( − x ) = x / (cid:18) J − / (cid:16) x / (cid:17) + J / (cid:16) x / (cid:17)(cid:19) , x > . (76)Ai( x ) is, up to normalization, the unique solution of the differential equationAi ′′ ( x ) = x Ai( x ) that is bounded for x ≥
0. Ai extends to an entire function.All zeros of Ai( x ) lie on the negative real axis; they will appear several timesbelow, and we denote them by a j = −| a j | , j = 1 , , . . . , with 0 < | a | < | a | , . . . .In other words, Ai( a j ) = Ai( −| a j | ) = 0 , j = 1 , , . . . (77)We have the asymptotic formula a j ∼ − (3 π/ / j / , which can be refined toan asymptotic expansion [1, 10.4.94], [17]. (Abramowitz and Stegun [1, 10.4.94]and Louchard [34] use a j as we do, while Tak´acs [44; 45; 46; 47; 48; 50] uses a j for | a j | ; Flajolet and Louchard [17] and Majumdar and Comtet [35] use α j for | a j | ;Darling [12] denotes our | a j | by σ j ; his α j are the zeros of J / ( x ) + J − / ( x ),and are equal to | a j | / in our notation, see (76).)We will later also need both the derivative Ai ′ and the integral of Ai. It seemsthat there is no standard notation for the latter, and we will useAI( x ) := Z ∞ x Ai( t ) d t = 13 − Z x Ai( t ) d t, (78)using R ∞ Ai( x ) d x = 1 /
13. Laplace transform
Let ψ ex ( t ) := E e − t B ex be the Laplace transform of B ex ; thus ψ ex ( − t ) := E e t B ex is the moment generating function of B ex . It follows from (53) that ψ ex ( t ) existsfor all complex t , and is thus an entire function, with ψ ex ( t ) = ∞ X k =0 E B k ex ( − t ) k k ! . (79)( ψ ex ( t ) is denoted φ ( t ) by Louchard [34], G (2 − / t ) by Flajolet and Louchard[17] and Ψ e ( t ) by Perman and Wellner [38].)Darling [12] found (in the context of (18)) the formula, with | a j | as in (77), ψ ex ( t ) = √ π t ∞ X j =1 exp (cid:0) − − / | a j | t / (cid:1) , t > x ≥ Z ∞ (cid:0) e − xt − (cid:1) ψ ex ( t / ) d t √ πt = 2 / (cid:18) Ai ′ (2 / x )Ai(2 / x ) − Ai ′ (0)Ai(0) (cid:19) , (81) . Janson/Brownian areas and Wright’s constants proved by Louchard [33]. We give a (related but somewhat different) proof of(81) in Appendix C.4. The relation (80) follows easily by Laplace inversion from(81) and the partial fraction expansion, see [17],Ai ′ ( z )Ai( z ) − Ai ′ (0)Ai(0) = ∞ X j =1 (cid:18) z − a j + 1 a j (cid:19) . (82)A proof of (80) by methods from mathematical physics is given by Majumdarand Comtet [35].Taking the derivative with respect to x in (81), we find, for x > Z ∞ e − xt ψ ex ( t / ) d t √ πt = − / dd x (cid:18) Ai ′ (2 / x )Ai(2 / x ) (cid:19) = (cid:18) / Ai ′ (2 / x )Ai(2 / x ) (cid:19) − x, (83)or, by the changes of variables x − / x , t / x , as in [34], Z ∞ e − xt ψ ex ( √ t / ) d t √ πt = − x (cid:18) Ai ′ ( x )Ai( x ) (cid:19) = 2 (cid:18) Ai ′ ( x )Ai( x ) (cid:19) − x. (84)
14. Series expansions for distribution and density functions
Let F ex ( x ) := P ( B ex ≤ x ) and f ex ( x ) := F ′ ( x ) be the distribution and densityfunctions of B ex . (Tak´acs [44; 45; 46; 47; 48] uses W and W ′ for these, while Fla-jolet and Louchard [17] use W and w for the distribution and density functionsof A = 2 / B ex ; thus their W ( x ) = F ex (2 − / x ) and w ( x ) = 2 − / f ex (2 − / x ).)Since B ex > x > a j denote the zeros of the Airy function. Darling [12] found byLaplace inversion from (80) (in our notation) F ex ( x ) = 2 √ π ∞ X j =1 | a j | − / p (cid:0) √ | a j | − / x (cid:1) , (85)where p is the density of the positive stable distribution with exponent 2 / − t / ); in the notation of Feller [15,Section XVII.6] p ( x ) = p (cid:0) x ; , − (cid:1) = 1 πx ∞ X k =1 ( − k − sin(2 kπ/
3) Γ(2 k/ k ! x − k/ . (86)By Feller [15, Lemma XVII.6.2] also p ( x ) = x − / p (cid:0) x − / ; , − (cid:1) , (87)where p ( x ; , − ) = πx P ∞ k =1 ( − k − sin(2 kπ/ Γ(2 k/ k ! x k is the density of aspectrally negative stable distribution with exponent 3 /
2. (Tak´acs [44] defines . Janson/Brownian areas and Wright’s constants the function g ( x ) = 2 − / p (2 − / x ; , − ); this is another such density functionwith a different normalization. Takacs [49] uses g for p .) The stable densityfunction p can also be expressed by a Whittaker function W or a confluenthypergeometric function U [1, 13.1], [32, 9.10 and 9.13.11] (where U is denotedΨ), p ( x ) = (cid:16) π (cid:17) / x − exp (cid:16) − x (cid:17) W , (cid:16) x (cid:17) (88)= 2 / / π / x − / exp (cid:16) − x (cid:17) U (cid:16) , , x (cid:17) . (89)Hence (85) can be written [44; 45; 46; 47; 48] F ex ( x ) = 2 / − / x − / ∞ X j =1 a j exp (cid:16) − | a j | x (cid:17) U (cid:16) ,
43 ; 2 | a j | x (cid:17) . (90)= √ x ∞ X j =1 v / j e − v j U (cid:16) ,
43 ; v j (cid:17) , (91)with v j = 2 | a j | / x , which leads to [17; 44; 45; 46; 47; 48] f ex ( x ) = 2 √ x ∞ X j =1 v / j e − v j U (cid:16) − ,
43 ; v j (cid:17) . (92)
15. Asymptotics of distribution and density functions
Louchard [34] gave the two first terms in an asymptotic expansion of f ex ( x ) andthe first term for F ex ( x ) as x →
0; it was observed by Flajolet and Louchard[17] that full asymptotic expansions readily follow from (92) (for f ex ; the resultfor F ex follows similarly from (90), or by integration); note that only the termwith j = 1 in (92) is significant as x → U given in e.g. [32, (9.12.3)]. The first terms are (correcting typos in [34] and[17]), as x → f ex ( x ) ∼ e − | a | x (cid:16) | a | / x − − | a | / x − − | a | − / x − + . . . (cid:17) , (93) F ex ( x ) ∼ e − | a | x (cid:16) | a | / x − + | a | − / − | a | − / x + . . . (cid:17) . (94)As in all other asymptotic expansions in this paper, we do not claim here thatthere is a convergent infinite series on the right hand side; the notation (using ∼ instead of =) signifies only that if we truncate the sum after an arbitraryfinite number of terms, the error is smaller order than the last term. (Hence,more precisely, the error is of the order of the first omitted term.) In fact, theseries in (93) and (94) diverge for every x > U does so. . Janson/Brownian areas and Wright’s constants For x → ∞ , it was observed by Cs¨org˝o, Shi and Yor [11] that (53) implies − ln P ( B ex > x ) = − ln (cid:0) − F ex ( x ) (cid:1) ∼ − x , x → ∞ . (95)Another proof was given by Fill and Janson [16]. Much more precise resultswere obtained by Janson and Louchard [25] who found, as x → ∞ , asymptoticexpansions beginning with f ex ( x ) ∼ √ √ π x e − x (cid:18) − x − − x − − x − + . . . (cid:19) , (96)1 − F ex ( x ) ∼ √ √ π xe − x (cid:18) − x − − x − − x − + . . . (cid:19) . (97)
16. Airy function expansions
Following Louchard [34], we expand the left hand side of (81) in an asymptoticseries as x → ∞ , using (79), and find after replacing 2 / x by x the asymptoticseries Ai ′ ( x )Ai( x ) ∼ − x / + ∞ X k =1 ( − k E B k ex Γ((3 k − / √ π k ! 2 k/ x (1 − k ) / . (98)Note that the infinite sum in (98) diverges by (53); cf. the discussion in Sec-tion 15. For the proof of (98), we thus cannot simply substitute (79) into (81);instead we substitute a truncated version (a finite Taylor series) ψ ex ( t ) = N X k =0 E B k ex ( − t ) k k ! + O ( t N +1 ) , (99)where N is finite but arbitrary. In similar situations for other Brownian areasin later sections, for example with (133) below for the Brownian bridge, thisyields directly a sum of Gamma integrals and an asymptotic expansion of thedesired type. In the present case, however, we have to work a little more to avoiddivergent integrals. One possibility is to substitute (99) into the differentiatedversion (84), evaluate the integrals, subtract x − / from both sides, and integratewith respect to x . Another possibility is to write the left hand side of (81) as Z ∞ (cid:0) e − xt − (cid:1) d t √ πt − Z ∞ (cid:0) ψ ex ( t / ) − (cid:1) d t √ πt + Z ∞ e − xt (cid:0) ψ ex ( t / ) − (cid:1) d t √ πt (100)and subsitute (99) into the third integral, noting that the first integral is −√ x / and the second is a constant (necessarily equal to 2 / Ai ′ (0) / Ai(0)). . Janson/Brownian areas and Wright’s constants Using (6), we can rewrite (98) asAi ′ ( x )Ai( x ) ∼ ∞ X k =0 ( − k K k x / − k/ , x → ∞ . (101)This is also, with a change of variables, given in Tak´acs [45]. Flajolet andLouchard [17] give this expansion with the coefficients ( − k Ω k / (2 k k !), whichis equivalent by (10). They give also related expansions involving Bessel andconfluent hypergeometric functions with coefficients Ω k /k !. An equivalent ex-pansion with coeficients 3( k − c k − , which equals Ω k /k ! by (41), was given byVobly˘ı [55], see [24, (8.14) and (8.15)].The Airy function and its derivative have, as x → ∞ , the asymptotic expan-sions [1, 10.4.59 and 10.4.61] (we write c ′ k and d ′ k instead of c k and d k to avoidconfusion with our c k and d k above)Ai( x ) ∼ √ π x − / e − x / / ∞ X k =0 ( − k c ′ k (cid:16) (cid:17) k x − k/ , (102)Ai ′ ( x ) ∼ − √ π x / e − x / / ∞ X k =0 ( − k d ′ k (cid:16) (cid:17) k x − k/ , (103)where c ′ k = Γ(3 k + )54 k k ! Γ( k + ) , d ′ k = − k + 16 k − c ′ k , k ≥ . (104)These asymptotic expansions can also be written using e k or α k , since, by (64),(71) and (104), 3 k c ′ k = e k , (105)(3 / k c ′ k = α k , (106)(3 / k d ′ k = α ′ k := − k + 16 k − α k . (107)We thus obtain from (101)–(103), after the change of variables z = − x − / ր ∞ X k =0 K k z k = − P ∞ k =0 (3 / k d ′ k z k P ∞ k =0 (3 / k c ′ k z k = − P ∞ k =0 α ′ k z k P ∞ k =0 α k z k . (108)These asymptotic expansions can also be written as hypergeometric series.We have, as is easily verified, the equalities for formal power series ∞ X k =0 c ′ k z k = F (cid:0) / , / z/ (cid:1) , (109) ∞ X k =0 d ′ k z k = F (cid:0) / , − / z/ (cid:1) . (110) . Janson/Brownian areas and Wright’s constants Hence, (102) and (103) can be written, for x → ∞ ,Ai( x ) ∼ √ π x − / e − x / / F (cid:16) ,
16 ; − x / (cid:17) , (111)Ai ′ ( x ) ∼ − √ π x / e − x / / F (cid:16) , −
16 ; − x / (cid:17) , (112)and (108) can be written ∞ X k =0 K k z k = − F (cid:0) / , − /
6; 3 z/ (cid:1) F (cid:0) / , /
6; 3 z/ (cid:1) . (113)Flajolet and Louchard [17] give the equivalent formula, see (10), ∞ X k =0 Ω k w k k ! = − F (cid:0) / , − /
6; 3 w/ (cid:1) F (cid:0) / , /
6; 3 w/ (cid:1) . (114)By the asymptotic expansion for Bessel functions [32, (5.11.10)] I ν ( x ) ∼ e x (2 πx ) − / F (1 / ν, / − ν ; 1 / x ) , x → ∞ , (115)this can, as in Flajolet and Louchard [17], also be written (with arbitrary signs) − I ± / ( x/ I ± / ( x/ ∼ ∞ X k =0 Ω k x − k k ! = ∞ X k =0 ω k x − k , x → ∞ . (116)The coefficients in this asymptotic series can be rewritten in various ways bythe relations in Sections 2 and 6, for example by (41) as 3( k − c k − for k ≥ e k =3 k c ′ k and α k = (3 / k c ′ k by (105) and (106).On the other hand, by differentiating (101) (which is allowed, e.g. becausethe asymptotic expansion holds in a sector in the complex plane), ∞ X k =0 ( − k (1 − k ) K k x − / − k/ ∼ Ai ′′ ( x )Ai( x ) − (cid:18) Ai ′ ( x )Ai( x ) (cid:19) ∼ x − ∞ X k =0 ( − k K k x / − k/ ! . (117)This gives the equation for formal power series ∞ X k =0 (3 k − K k z k +1 = 1 − ∞ X k =0 K k z k ! , (118)which is equivalent to the quadratic recursion (7). . Janson/Brownian areas and Wright’s constants
17. A continued fraction
As noted in [24, (8.16)], (116) yields (after substituting z = 1 /x ) the asymptoticexpansion2 z + 18 z + z + z + 126 z + ... ∼ − ∞ X k =0 Ω k z k k ! = − ∞ X k =0 ω k z k , z → , (119)where the continued fraction has denominators (6 n + 2) z .
18. Moments and Airy zeroes
Define, following Flajolet and Louchard [17],Λ( s ) := ∞ X j =1 | a j | − s , Re s > / . (120)Since | a j | ∼ (3 πj/ / [1, 10.4.94], the sum converges and Λ is analytic forRe s > /
2. (Flajolet and Louchard [17] call Λ the root zeta function of the Airyfunction Ai.)By (80), for Re s > Z ∞ t s − ψ ex ( t ) d t = √ π Z ∞ t s +1 ∞ X j =1 exp (cid:0) − − / | a j | t / (cid:1) d tt = 3 √ π ∞ X j =1 Z ∞ u (3 s +3) / exp (cid:0) − − / | a j | u (cid:1) d uu = 3 √ π ( s +1) / ∞ X j =1 | a j | − (3 s +3) / Γ (cid:16) s + 32 (cid:17) = 3 √ π s/ Γ (cid:16) s + 32 (cid:17) Λ (cid:16) s + 32 (cid:17) . (121)In particular, this is finite for all s >
0, and thus Lemma B.1 in Appendix Bapplies and shows that B ex has negative moments of all orders. Since all (posi-tive) moments too are finite, E B s ex is an entire function of s , and by (121) and(335), Λ extends to a meromorphic function in C and, as shown by Flajolet andLouchard [17], E B s ex = 3 √ π − s/ Γ( − s ) Γ (cid:16) − s (cid:17) Λ (cid:16) − s (cid:17) , s ∈ C . (122)Since the left hand side has no poles, while the Gamma factors are non-zero,Λ (cid:0) (3 − s ) / (cid:1) can have poles only at the poles of Γ( − s ), i.e. at s = 0 , , . . . ;hence, Λ( z ) can have poles only at z = , 0, − , − − , . . . . Moreover, E B s ex . Janson/Brownian areas and Wright’s constants is non-zero for real s ; hence Γ (cid:0) (3 − s ) / (cid:1) Λ (cid:0) (3 − s ) / (cid:1) and Γ( − s ) have exactlythe same zeros (i.e., none) and the same poles. Consequently, Λ( z ) has simplepoles at z = , − , − , . . . , but is finite and non-zero at 0, − −
6, . . . (whereΓ has a pole); furthermore, Λ is zero at the other poles of Γ, i.e. at − − − − −
7, . . . . This was shown by Flajolet and Louchard [17] (by a partlydifferent approach, extending Λ( s ) by explicit formulas).As shown in Flajolet and Louchard [17], as a simple consequence of (82), for | z | < | a | , Ai ′ ( z )Ai( z ) − Ai ′ (0)Ai(0) = X k ≥ Λ( k + 1)( − z ) k , (123)and thus the values of Λ( s ) at positive integers s = 2 , , . . . can be computedfrom the Taylor series of Ai, given for example in [1, 10.4.2–5]. This and (122)gives explicit formulas for the negative moments B − s ex when s is an odd multipleof 1 /
3, including when s is an odd integer; see Flajolet and Louchard [17].Alternatively, these formulas follow from (340) in Appendix B. For example, E B − / = 2 / / π / Γ(1 / = 2 − / / π − / Γ(2 / ≈ . , (124) E B − = 3 √ π (cid:18) − / Γ(2 / π (cid:19) ≈ . , (125)see further Section 29. We have here used the standard formulaΓ(1 / /
3) = π sin( π/
3) = 2 π √ . (126)Note also that if − < Re s < − /
2, then by (81), Fubini’s theorem and(335) in Appendix B,2 / Z ∞ x s − (cid:18) Ai ′ (2 / x )Ai(2 / x ) − Ai ′ (0)Ai(0) (cid:19) d x = Z ∞ Z ∞ x s − (cid:0) e − xt − (cid:1) ψ ex ( t / ) d t √ πt d x = Γ( s ) Z ∞ t − s ψ ex ( t / ) d t √ πt = Γ( s ) √ π · Z ∞ u − s/ − / ψ ex ( u ) d uu = 2Γ( s )3 √ π Γ (cid:16) − s + 13 (cid:17) E B (2 s +1) / . (127)By (122), this equals2 / − s/ Γ( s )Γ(1 − s )Λ(1 − s ) = 2 (1 − s ) / π sin( πs ) Λ(1 − s ) , (128)as shown directly by Flajolet and Louchard [17]. . Janson/Brownian areas and Wright’s constants
19. Integrals of powers of B ex Several related results are known for other functionals of a Brownian excursionor other variants of a Brownian motion. We describe some of them in this andthe following sections, emphasizing the similarities with the results above, andin particular linear and quadratic recurrencies for moments and related formulasfor generating functions.The results above for moments of B ex = R B ex ( t ) d t have been generalizedto joint moments of the integrals R B ex ( t ) ℓ d t , ℓ = 1 , , . . . , by Nguyˆen Thˆe [37](mainly ℓ = 1 and 2) and Richard [41] (all ℓ ≥ B ex ( ℓ = 1), then A r,s in Nguyˆen Thˆe[37] specializes to A r, = 2 r K r = σ r − , r ≥
0; (129)similarly, in Richard [41], f k, ,..., = 2 − k K k , k ≥ , (130) m k, ,..., = 2 k/ E B k ex = E A k , k ≥ , (131)so X ,n d −→ A , and η k, ,..., = E B k ex . The recursions in both [37] and [41]specialize to (7) (or an equivalent quadratic recursion).Janson [21] has given further similar results, including a similar quadraticrecursion with double indices, for (joint) moments of B ex and another functionalof a Brownian excursion; see also Chassaing and Janson [23].
20. Brownian bridge
Let B br := R | B br ( t ) | d t , the integral (or average) of the absolute value of aBrownian bridge. (Shepp [42] and Rice [40] use ξ for B br ; Johnson and Killeen[28] use L ; Tak´acs [47] uses σ . Cifarelli [9] uses C ( x ) for the distribution functionof 2 / B br .) Comtet, Desbois and Texier [10] give also the alternative represen-tation B br := Z | B br ( t ) | d t d = Z | B br ( t ) − B br (1 − t ) | d t, (132)which follows because B br ( t ) − B br (1 − t ) d = B br (2 t ), t ∈ [0 , / ψ br ( t ) := E e − t B br . (See Appendix C.2 for a proof.) An equivalent re-sult, in physical formulation, was proved by path integral methods by Altshuler,Aronov and Khmelnitsky [3], see Comtet, Desbois and Texier [10]. Shepp’s ver-sion is (with ψ br denoted φ by Shepp), cf. (81): Z ∞ e − xt ψ br ( √ t / ) d t √ t = −√ π Ai( x )Ai ′ ( x ) , x ≥ , (133) . Janson/Brownian areas and Wright’s constants while Cifarelli’s version, which actually is stated using the Bessel function K / ,cf. (74), can be written Z ∞ e − xt ψ br ( ut / ) d t √ πt = − (2 u ) − / Ai(2 / u − / x )Ai ′ (2 / u − / x ) , x ≥ , (134)for arbitrary u >
0. Clearly, (133) is the special case u = √ B br the are obtained by asymptotic expansions, cf. Section 16; Cifarelli [9]considers (134) as u → x → ∞ ; these areobviously equivalent. Introduce D n defined by E B n br = √ π − n/ n !Γ((3 n + 1) / D n , n ≥ . (135)(Tak´acs [47] denotes E B n br by M ∗ n ; Perman and Wellner [38] use L n for our D n , A for B br and µ n for E B n br ; Nguyˆen Thˆe [36] denotes E B n me by M Pn and uses Q n for our D n and a Pn for 2 − / − n/ D n .) We then obtain from (133), cf. (98)and the discussion after it, − Ai( x )Ai ′ ( x ) ∼ ∞ X k =0 ( − k D k x − k/ − / , x → ∞ , (136)and thus by (102), (103), (106), (107), cf. (108), as formal power series, ∞ X k =0 D k z k = P ∞ k =0 (3 / k c ′ k z k P ∞ k =0 (3 / k d ′ k z k = P ∞ k =0 α k z k P ∞ k =0 α ′ k z k . (137)Cifarelli [9] uses m k := 2 k/ E B k br = E (cid:0) / B br (cid:1) k (138)and writes the result of the expansion as m k = √ π k !Γ (cid:0) (3 k + 1) / (cid:1) ( − k C k , k ≥ , (139)with, as formal power series (we use a ′ k and b ′ k for a k and b k in [9]) ∞ X k =0 C k z k = P ∞ k =0 a ′ k z k P ∞ k =0 b ′ k z k , where a ′ k = ( − k c ′ k and b ′ k = ( − k d ′ k ; (140)this is by (138) equivalent to (135) and (137) together with C k = ( − k D k , k ≥ . (141)Cifarelli [9] gives, instead of a recursion relation, the solution to (140) by theformula ( − k C k = k X p =0 ( − k − p a ′ k − p | B | p = k X p =0 | a ′ k − p || B | p , (142) . Janson/Brownian areas and Wright’s constants where | B | p is given by the general determinant formula | B | k := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b ′ b ′ . . . b ′ k − b ′ k b ′ . . . b ′ k − b ′ k − . . . b ′ k − b ′ k − ... ... . . . ... ...0 0 . . . b ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ; (143)this is a general way (when b ′ = 1) to write the solution to ∞ X k =0 | B | k x k = ∞ X k =0 ( − k b ′ k x k ! − . (144)Shepp [42] gives, omitting intermediate steps, the result of expanding (133)as (using ξ for B br ) E B n br = √ π n/ n Γ (cid:0) (3 n + 1) / (cid:1) ¯ e n , n ≥ , (145)where ¯ e n satisfies the linear recursion¯ e n = Γ(3 n + )Γ( n + ) + n X k =1 ¯ e n − k (cid:18) nk (cid:19) k + 16 k − k + )Γ( k + ) , n ≥ . (146)(Shepp writes e n but we use ¯ e n to avoid confusion with e r in (64).)We have, e.g. by (145), (135) and (141),¯ e n = 36 n n ! D n = ( − n n ! C n , n ≥ . (147)Hence, (146) can, as in Perman and Wellner [38] (where further, as said above, α n is denoted by γ n ), be rewritten as D n = α n − n X i =1 α ′ i D n − i = α n + n X i =1 i + 16 i − α i D n − i , n ≥ , (148)which also follows from (137).Tak´acs [47] found, by different methods, a quadratic recurrence analogous to(6) and (7), namely (135) together with D = 1 and D n = 3 n − D n − − n − X i =1 D i D n − i , n ≥ . (149)This follows also by differentiating (136), which yields, as x → ∞ , ∞ X k =0 ( − k +1 k + 12 D k x − k/ − / ∼ − x )Ai ′′ ( x )Ai ′ ( x ) = − x (cid:16) Ai( x )Ai ′ ( x ) (cid:17) . Janson/Brownian areas and Wright’s constants ∼ − ∞ X k =0 ( − k D k x − k/ ! (150)and thus 1 + ∞ X k =0 k + 12 D k z k +1 = ∞ X k =0 D k z k ! . (151)Note further that (101) and (136) show that ∞ X k =0 D k z k = (cid:18) − ∞ X k =0 K k z k (cid:19) − = K P ∞ k =0 K k z k (152)and thus k X i =0 D i K k − i = 0 , k ≥ , (153)which can be regarded as a linear recursion for D n given K n , or conversely.Explicitly, D n = 2 n − X i =0 D i K n − i , n ≥ . (154)It follows easily from the asymptotics (58) that the term with i = 0 in (154)asymptotically dominates the sum of the others and thus D n ∼ K n ∼ π (cid:16) (cid:17) n ( n − ∼ r πn (cid:16) n e (cid:17) n as n → ∞ , (155)as given by Tak´acs [47] (derived by him from (149) without further details).Consequently, cf. (53), by (135), or alternatively by Tolmatz [51] or Janson andLouchard [25], E B n br ∼ √ (cid:16) n e (cid:17) n/ as n → ∞ . (156)The entire function Ai ′ has all its zeros on the negative real axis (just asAi, see Section 12), and we denote them by a ′ j = −| a ′ j | , j = 1 , , . . . , with0 < | a ′ | < | a ′ | < . . . ; thus, cf. (77),Ai ′ ( a ′ j ) = Ai ′ ( −| a ′ j | ) = 0 , j = 1 , , . . . (157)(Abramowitz and Stegun [1, 10.4.95], Rice [40] and Johnson and Killeen [28]use a ′ j as we do; Tak´acs [47; 49] uses a ′ j for our | a ′ j | ; Flajolet and Louchard [17]use α j for | a ′ j | ; Kac [29] and Johnson and Killeen [28] use δ j for 2 − / | a ′ j | .)The meromorphic function Ai( z ) / Ai ′ ( z ) has the residue Ai( a ′ j ) / Ai ′′ ( a ′ j ) =1 /a ′ j at a ′ j , and there is a convergent partial fraction expansionAi( z )Ai ′ ( z ) = ∞ X j =1 a ′ j ( z − a ′ j ) = − ∞ X j =1 | a ′ j | ( z + | a ′ j | ) , z ∈ C . (158) . Janson/Brownian areas and Wright’s constants Inversion of the Laplace transform in (133) yields, as found by Rice [40], ψ br ( t ) = 2 − / π / ∞ X j =1 | a ′ j | − t / e − − / | a ′ j | t / , t > . (159)Johnson and Killeen [28] found by a second Laplace transform inversion thedistribution function F br ( x ) := P ( B br ≤ x ), here written as in Tak´acs [47] with u j := | a ′ j | / (27 x ), F br ( x ) = 18 − / π / x − ∞ X j =1 e − u j u − / j Ai (cid:0) (3 u j / / (cid:1) = 2 − / / π / x − / ∞ X j =1 | a ′ j | e −| a ′ j | / (27 x ) Ai (cid:0) − / | a ′ j | x − / (cid:1) . (160)Numerical values are given by Johnson and Killeen [28] and Tak´acs [47].The density function f br = F ′ br can be obtained by termwise differentiationof (160), although no-one seems to have bothered to write it out explicitly;numerical values (obtained from (159)) are given by Rice [40].Asymptotic expansions of f br ( x ) and F br ( x ) as x → j = 1 in (160) is significant and the first terms of theasymptotic expansions are, as x → f br ( x ) ∼ e − | a ′ | x (cid:16) | a ′ | / x − − | a ′ | − / x − + | a ′ | − / x + O (cid:0) x (cid:1)(cid:17) , (161) F br ( x ) ∼ e − | a ′ | x (cid:16) | a ′ | − / − | a ′ | − / x + | a ′ | − / x + O (cid:0) x (cid:1)(cid:17) . (162)For x → ∞ we have by Tolmatz [51] and Janson and Louchard [25] asymptoticexpansions beginning with f br ( x ) ∼ √ √ π e − x (cid:18) x − + 1432 x − + O (cid:0) x − (cid:1)(cid:19) , (163)1 − F br ( x ) ∼ √ π x − e − x (cid:18) − x − + 1108 x − + O (cid:0) x − (cid:1)(cid:19) . (164)Nguyˆen Thˆe [37] extended some of these results to the joint Laplace trans-form and moments of B br and R | B br | d t . Specializing to B br only, his momentformula is E B r br = β r, := √ π − r/ r !Γ((3 k + 1) / B r, , r ≥ , (165)with (correcting several typos in [37]) B , = 1 and B r, = 3 r − B r − , − k − X i =1 B i, B r − i, , r ≥ . (166) . Janson/Brownian areas and Wright’s constants E B = 1 E B br = 14 q π E B = 760 E B = 21512 q π E B = 19720 m = 1 m = 12 √ π m = 1415 m = 2132 √ π m = 7645¯ e = 1 ¯ e = 9 ¯ e = 567 ¯ e = 91854 ¯ e = 28796229 D = 1 D = 14 D = 732 D = 2164 D = 14632048 C = 1 C = − C = 78 C = − C = 1463128¯ e ∗ = 1 ¯ e ∗ = 1 ¯ e ∗ = 7 ¯ e ∗ = 126 ¯ e ∗ = 4389 D ∗ = 1 D ∗ = 2 D ∗ = 14 D ∗ = 168 D ∗ = 2926 Table 2
Some numerical values for the Brownian bridge.
Clearly, these results are equivalent to (135) and (149), with B r, = 2 r D r = ( − r C r , r ≥ . (167)Nguyˆen Thˆe [37] gives also the relation r X i =0 A i, B r − i, = 0 , r ≥ , (168)which by (129) and (167) is equivalent to (153). Equivalently, ∞ X k =0 A k, y k = A , P ∞ k =0 B k, y k . (169)Some numerical values are given in Table 2; see further Cifarelli [9] ( m n ),Shepp [42] ( E B n br , ¯ e n ), Tak´acs [47] ( E B n br , D n ), Nguyˆen Thˆe [37] ( E B n br , B n,m ).We define D ∗ n := 2 n D n and ¯ e ∗ n := 9 − n ¯ e n = 4 n n ! D n (considered by Shepp [42]);note that these are integers.
21. Brownian motion
Let B bm := R | B ( t ) | d t , the integral of the absolute value of a Brownian motionon the unit interval, and let ψ bm ( t ) := E e − t B bm be its Laplace transform. Kac[29] gave the formula ψ bm ( t ) = ∞ X j =1 κ j e − δ j t / , t > , (170)where δ j = 2 − / | a ′ j | and, with Kac’s notation P ( y ) = 2 / Ai (cid:0) − / y (cid:1) , κ j := 1 + 3 R δ j P ( y ) d y δ j P ( δ j ) = 1 + 3 R a ′ j Ai( x ) d x | a ′ j | Ai( a ′ j ) . (171) . Janson/Brownian areas and Wright’s constants (Tak´acs [49] uses C j for κ j .) Using the function AI defined in (78), (171) canbe written κ j = AI( a ′ j ) | a ′ j | Ai( a ′ j ) . (172)In fact, see Tak´acs [49] and Perman and Wellner [38], or (362) in Appendix C.1below, Z ∞ e − xt ψ bm (cid:0) √ t / (cid:1) d t = − AI( x )Ai ′ ( x ) , x > . (173)The meromorphic function − AI( z ) / Ai ′ ( z ) has residue − AI( a ′ j ) / Ai ′′ ( a ′ j ) = κ j at a ′ j , and the partial fraction expansion − AI( z )Ai ′ ( z ) = ∞ X j =1 κ j z − a ′ j = ∞ X j =1 κ j z + | a ′ j | , z ∈ C . (174)An inversion of the Laplace transform in (173) thus yields (170).Tak´acs [49] found by a second Laplace transform inversion, this time of (170),the density function f bm ( x ) (by him denoted h ( x )) of B bm as f bm ( x ) = ∞ X j =1 κ j √ | a ′ j | − / p (cid:0) √ | a ′ j | − / x (cid:1) (175)= √ √ π x ∞ X j =1 κ j ( v ′ j ) / e − v ′ j U (cid:16) ,
43 ; v ′ j (cid:17) , (176)where p is the stable density in (87), and v ′ j = 2 | a ′ j | / x (denoted v j by Tak´acs[49]), cf. (85), (91), (92).Asymptotic expansions of f bm ( x ) and the distribution function F bm ( x ) as x → U given in e.g. [32,(9.12.3)], cf. Section 15; only the term with j = 1 in (176) is significant and thefirst terms of the asymptotic expansions are, as x → f bm ( x ) ∼ κ √ π e − | a ′ | x (cid:16) | a ′ | / x − + | a ′ | − / − | a ′ | − / x + . . . (cid:17) , (177) F bm ( x ) ∼ κ √ π e − | a ′ | x (cid:16) | a ′ | − / x − | a ′ | − / x + | a ′ | − / x + . . . (cid:17) . (178)For x → ∞ we have by Tolmatz [52] and Janson and Louchard [25] asymptoticexpansions beginning with f bm ( x ) ∼ √ √ π e − x / (cid:18) x − − x − + . . . (cid:19) , (179)1 − F bm ( x ) ∼ √ √ π x − e − x / (cid:18) − x − + 2281 x − + . . . (cid:19) . (180) . Janson/Brownian areas and Wright’s constants Tak´acs [49] also found recursion formulas for the moments. Define L n by E B n bm = 2 − n/ n !Γ((3 n + 2) / L n , n ≥ . (181)(Tak´acs [49] denotes E B n bm by µ n ; Perman and Wellner [38] use K n for our L n , A for B bm and ν n for E B n bm ; Nguyˆen Thˆe [36] uses a Bn for 2 − n/ L n .) Anasymptotic expansion of the left hand side of (173) yields, arguing as for (98)and (136), − AI( x )Ai ′ ( x ) ∼ ∞ X k =0 ( − k L k x − k/ − , x → ∞ . (182)Recall the asymptotic expansions (102) and (103) of Ai( x ) and Ai ′ ( x ); there isa similar expansion of AI( x ) [49], see Appendix A,AI( x ) ∼ √ π x − / e − x / / ∞ X k =0 ( − k β k x − k/ , x → ∞ , (183)where β k , k ≥
0, are given by β = 1 and the recursion relation (obtained fromcomparing a formal differentiation of (183) with (102)) β k = α k + 3(2 k − β k − , k ≥ . (184)(Tak´acs [49] further uses h k = (2 / k β k .) Hence, (182) yields, using (103) and(107), the equality (for formal power series) ∞ X k =0 L k z k = P ∞ k =0 β k z k P ∞ k =0 (3 / k d ′ k z k = P ∞ k =0 β k z k P ∞ k =0 α ′ k z k . (185)Multiplying by the denominator and identifying coefficients leads to the recur-sion by Tak´acs [49], L n = β n + n X j =1 j + 16 j − α j L n − j , n ≥ . (186)Differentiation of (182) yields, using Ai ′′ ( x ) = x Ai( x ),Ai( x )Ai ′ ( x ) + x Ai( x )AI( x )Ai ′ ( x ) ∼ ∞ X k =0 ( − k +1 k + 22 L k x − k/ − , x → ∞ , (187)and thus by (136) and (182) the equality − ∞ X k =0 D k z k + ∞ X k =0 D k z k ∞ X k =0 L k z k = ∞ X k =0 k + 22 L k z k +1 = ∞ X k =1 k − L k − z k , (188) . Janson/Brownian areas and Wright’s constants E B = 1 E B bm = 23 r π E B = 38 E B = 263630 r π E B = 9032560 β = 1 β = 4148 β = 92414608 β = 5075225663552 β = 5153008945127401984 L = 1 L = 1 L = 94 L = 26332 L = 270964 L ∗ = 1 L ∗ = 8 L ∗ = 144 L ∗ = 4208 L ∗ = 173376 Table 3
Some numerical values for the Brownian motion. which is equivalent to L = 1 and the recursion formula3 n − L n − = n X j =1 D n − j L j , n ≥ L n = 3 n − L n − − n − X j =1 D n − j L j , n ≥ . (190)Tak´acs [49] proves also the asymptotic relations, see further Tolmatz [52] andJanson and Louchard [25], E B n bm ∼ √ (cid:16) n e (cid:17) n/ , (191) L n ∼ √ (cid:16) n e (cid:17) n . (192)Some numerical values are given in Table 3; see further Tak´acs [49] andPerman and Wellner [38]. We define L ∗ n := 2 n L n ; these are integers by (190).
22. Brownian meander
Let B me := R | B me ( t ) | d t , the integral of a Brownian meander on the unitinterval, and let ψ me ( t ) := E e − t B me be its Laplace transform.Tak´acs [50] and Perman and Wellner [38] give formulas equivalent to Z ∞ e − xt ψ me (cid:0) √ t / (cid:1) d t √ πt = AI( x )Ai( x ) , x >
0; (193)see also Appendix C.3.Define Q n by E B n me = √ π − n/ n !Γ((3 n + 1) / Q n , n ≥ . (194)(Tak´acs [50] denotes E B n me by M n and ψ me by Ψ; Perman and Wellner [38] use R n for our Q n , A mean for B me and ρ n for E B n me ; Nguyˆen Thˆe [36] denotes E B n me . Janson/Brownian areas and Wright’s constants by M Fn and uses a Fn for 2 / − n/ Q n ; Majumdar and Comtet [35] denote E B n me by a n and Q n by R n .) An asymptotic expansion of the left hand side of (193)yields, arguing as for (98), (136) and (182),AI( x )Ai( x ) ∼ ∞ X k =0 ( − k Q k x − k/ − / , x → ∞ . (195)The asymptotic expansions (102) and (183) of Ai( x ) and AI( x ) yield, using(106), the equality (for formal power series) ∞ X k =0 Q k z k = P ∞ k =0 β k z k P ∞ k =0 (3 / k c ′ k z k = P ∞ k =0 β k z k P ∞ k =0 α k z k . (196)Differentiation of (195) yields − − Ai ′ ( x )AI( x )Ai( x ) ∼ ∞ X k =0 ( − k +1 k + 12 Q k x − k/ − / , x → ∞ , (197)and thus, using (101), the equality ∞ X k =0 k + 12 Q k z k +1 = − − ∞ X k =0 K k z k ∞ X k =0 Q k z k , (198)which is equivalent to the recursion formula Q n = 3 n − Q n − + 2 n X j =1 K j Q n − j , n ≥ . (199)with Q = 1, proved by Tak´acs [50] by a different method, and the differentialequation 3 z dd z ∞ X k =0 Q k z k + (cid:16) z + 4 ∞ X k =0 K k z k (cid:17) ∞ X k =0 Q k z k = − , (200)also given by Tak´acs [50] (in a slightly different form, with − z substitutedfor z ). On the other hand, multiplying by P ∞ k =0 α k z k in (196) and identifyingcoefficients leads to another recursion by Tak´acs [50] and Perman and Wellner[38]: Q n = β n − n X j =1 α j Q n − j , n ≥ . (201)Furthermore, (108), (185) and (196) yield the equality − ∞ X k =0 K k z k ∞ X k =0 L k z k = ∞ X k =0 Q k z k , (202) . Janson/Brownian areas and Wright’s constants E B = 1 E B me = 34 q π E B = 5960 E B = 465512 q π E B = 53453696 Q = 1 Q = 34 Q = 5932 Q = 46564 Q = 801752048 Q ∗ = 1 Q ∗ = 6 Q ∗ = 118 Q ∗ = 3720 Q ∗ = 160350 Table 4
Some numerical values for the Brownian meander. and thus the relation Q n = − n X j =0 K j L n − j , n ≥ . (203)By (152) we obtain from (202) also ∞ X k =0 D k z k ∞ X k =0 Q k z k = ∞ X k =0 L k z k , (204)and thus the relation L n = n X j =0 D j Q n − j , n ≥ . (205)Tak´acs [50] proves the asymptotic relations, see also Janson and Louchard[25], E B n me ∼ √ πn (cid:16) n e (cid:17) n/ , (206) Q n ∼ √ (cid:16) n e (cid:17) n . (207)Some numerical values are given in Table 4; see further Tak´acs [50]. We define Q ∗ n := 2 n Q n ; these are integers by (203) (or by (199) and (12)).The meromorphic function AI( z ) / Ai( z ) has residue r j := AI( a j ) / Ai ′ ( a j ) at a j , and the partial fraction expansionAI( z )Ai( z ) = ∞ X j =1 r j z + | a j | = ∞ X j =1 r j (cid:16) z + | a j | − | a j | (cid:17) + 13Ai(0) , z ∈ C , (208)where the first sum is conditionally convergent only and the second sum is abso-lutely convergent, since | a j | ≍ j / and | r j | ≍ j − / . (Note that r j alternates insign.) In fact, the second version follows from an absolutely convergent partialfraction expansion of AI( z ) / ( z Ai( z )), using AI(0) = 1 /
3, and the first then fol-lows easily. A Laplace transform inversion (considering, for example, the Laplacetransform of √ t ψ me ( √ t / )) yields the formula by Tak´acs [50] ψ me ( t ) = 2 − / t / √ π ∞ X j =1 r j e − − / | a j | t / , t > . (209) . Janson/Brownian areas and Wright’s constants A second Laplace inversion, see Tak´acs [50], yields the distribution function F me ( x ) of B me as, using u j = | a j | / (27 x ) and R j = | a j | r j = | a j | AI( a j ) / Ai ′ ( a j ), F me ( x ) = 18 − / π / x − ∞ X j =1 R j e − u j u − / j Ai (cid:0) (3 u j / / (cid:1) = 2 − / / π / x − / ∞ X j =1 r j e −| a j | / (27 x ) Ai (cid:0) − / a j x − / (cid:1) . (210)Numerical values are given by Tak´acs [50]. The density function f me ( x ) can beobtained by termwise differentiation.A physical proof of (209) by path integral methods is given by Majumdarand Comtet [35] (denoting our r j by B ( α j )), who then conversely derive (193)from (209).Asymptotic expansions of f me ( x ) and F me ( x ) as x → j = 1 in (210) is significant and the first terms of theasymptotic expansions are, as x → f me ( x ) ∼ r e − | a | x (cid:16) | a | / x − − | a | − / x − + | a | − / x + O (cid:0) x (cid:1)(cid:17) , (211) F me ( x ) ∼ r e − | a | x (cid:16) | a | − / − | a | − / x + | a | − / x + O (cid:0) x (cid:1)(cid:17) . (212)For x → ∞ , Janson and Louchard [25] found asymptotic expansions begin-ning with f me ( x ) ∼ √ xe − x / (cid:18) − x − − x − + . . . (cid:19) , (213)1 − F me ( x ) ∼ √ e − x / (cid:18) − x − + 5162 x − + . . . (cid:19) . (214)
23. Brownian double meander
Define B dm ( t ) := B ( t ) − min ≤ u ≤ B ( u ) , t ∈ [0 , . (215)This is a non-negative continuous stochastic process on [0 ,
1] that a.s. is 0 at aunique point τ ∈ [0 ,
1] (the time of the minimum of B ( t ) on [0 , τ has an arcsine (= Beta( , )) distribution with density π − ( t (1 − t )) − / ; moreover, given τ , the processes B dm ( τ − t ), t ∈ [0 , τ ], and B dm ( τ + t ), t ∈ [0 , − τ ], are two independent Brownian meanders on the respective intervals,see Williams [57], Denisov [13] and Bertoin, Pitman and Ruiz de Chavez [6].(This may be seen as a limit of an elementary corresponding result for simplerandom walks.) We therefore call B dm a Brownian double meander . . Janson/Brownian areas and Wright’s constants Let B dm := R B dm ( t ) d t be the Brownian double meander area. By thedefinition, we have the formula B dm := Z B dm ( t ) d t = Z B ( t ) d t − min ≤ t ≤ B ( t ); (216)note the analogy with (16) for B ex and B br . As in Section 3, there are furtherinteresting equivalent forms of this. Since B is symmetric, B d = − B , we alsohave B dm d = max ≤ t ≤ B ( t ) − Z B ( t ) d t = max ≤ t ≤ B ( t ) , (217)where B ( t ) := B ( t ) − Z B ( s ) d s, ≤ t ≤ , (218)is a continuous Gaussian process with mean 0 and integral identically zero; itscovariance function is, by straightforward calculation, given byCov (cid:0) B ( s ) , B ( t ) (cid:1) = − max( s, t ) + ( s + t ) , s, t ∈ [0 , . (219)Let ψ dm ( t ) := E e − t B dm be the Laplace transform of B dm . Majumdar andComtet [35] give the formula (using C ( α j ) for our r j ), where r j = AI( a j ) / Ai ′ ( a j )as in Section 22, ψ dm ( t ) = 2 − / t / ∞ X j =1 r j e − − / | a j | t / , t >
0; (220)they then derive from it Z ∞ e − xt ψ dm (cid:0) √ t / (cid:1) d t = (cid:18) AI( x )Ai( x ) (cid:19) , x > , (221)using the partial fraction expansion, cf. (208), (cid:18) AI( z )Ai( z ) (cid:19) = ∞ X j =1 r j ( z − a j ) = ∞ X j =1 r j ( z + | a j | ) , z ∈ C . (222)(There are only quadratic terms in this partial fraction expansion, becauseAI ′ ( z ) = − Ai( z ) and Ai ′′ ( z ) = z Ai( z ) vanish at the zeros of Ai; hence thereis no constant term in the expansion of AI( z ) / Ai( z ) at a pole a j .) Conversely,we will prove (221) by other methods in Section 27, and then (220) follows by(222) and a Laplace transform inversion.Majumdar and Comtet [35] found, by a Laplace transform inversion of (220),the density function f dm ( x ) of B dm as f dm ( x ) = 2 − / √ π x − / ∞ X j =1 | a j | r j e − v j (cid:16) U (cid:16) ,
43 ; v j (cid:17) + 2 U (cid:16) − ,
43 ; v j (cid:17)(cid:17) (223) . Janson/Brownian areas and Wright’s constants where U is the confluent hypergeometric function and v j = 2 | a j | / x , cf. (91)and (92).Asymptotic expansions of f dm ( x ) and the distribution function F dm ( x ) as x → U [32, (9.12.3)]; only the term with j = 1 in (223) is significant and the first termsof the asymptotic expansions are, as x → f dm ( x ) ∼ q π r e − | a | x (cid:16) | a | / x − − | a | / x − + | a | − / + . . . (cid:17) , (224) F dm ( x ) ∼ q π r e − | a | x (cid:16) | a | / x − + | a | − / x − | a | − / x + . . . (cid:17) , (225)For x → ∞ , Janson and Louchard [25] found asymptotic expansions begin-ning with f dm ( x ) ∼ √ √ π e − x / (cid:18) x − + 118 x − + . . . (cid:19) , (226)1 − F dm ( x ) ∼ √ √ π x − e − x / (cid:18) − x − + 29 x − + . . . (cid:19) . (227)The weaker statement − ln P ( B dm > x ) = − ln (cid:0) − F dm ( x ) (cid:1) ∼ − x / , as x → ∞ , (228)had earlier been proved by Majumdar and Comtet [35] using moment asymp-totics, see (235) below.Define W n by, in analogy with (181), E B n dm = 2 − n/ n !Γ((3 n + 2) / W n , n ≥ . (229)An asymptotic expansion of the left hand side of (221) yields, arguing as for(98), (136), (182) and (195), (cid:18) AI( x )Ai ′ ( x ) (cid:19) ∼ ∞ X k =0 ( − k W k x − k/ − , x → ∞ . (230)By (183), (102), (106) and (196), this yields the equality (for formal powerseries) ∞ X k =0 W k z k = (cid:18) P ∞ k =0 β k z k P ∞ k =0 (3 / k c ′ k z k (cid:19) = (cid:18) P ∞ k =0 β k z k P ∞ k =0 α k z k (cid:19) = ∞ X k =0 Q k z k ! . (231)Consequently, W n = n X j =0 Q j Q n − j , n ≥ , (232) . Janson/Brownian areas and Wright’s constants E B = 1 E B dm = r π E B = 1724 E B = 123140 r π E B = 29633840 W = 1 W = 32 W = 174 W = 110764 W = 296332 W ∗ = 1 W ∗ = 12 W ∗ = 272 W ∗ = 8856 W ∗ = 379264 Table 5
Some numerical values for the Brownian double meander. or, as given by Majumdar and Comtet [35] (where E B n dm is denoted by µ n ), E B n dm = 1 π n X m =0 (cid:18) nm (cid:19) B (cid:16) m + 12 , n − m ) + 12 (cid:17) E B m me E B n − m me . (233)From (232) and (207) follows the asymptotic relation W n ∼ Q Q n = 2 Q n ∼ √ (cid:16) n e (cid:17) n , (234)and thus by (229), see also Janson and Louchard [25], E B n dm ∼ √ (cid:16) n e (cid:17) n/ . (235)Some numerical values are given in Table 5; see also Majumdar and Comtet[35]. We define W ∗ n := 2 n W n ; these are integers by (232).
24. Positive part of a Brownian bridge
Let x + := max( x,
0) and x − := ( − x ) + and define B ± br := Z B br ( t ) ± d t ; (236)thus B +br is the average of the positive part and B − br the average of the nega-tive part of a Brownian bridge. (Perman and Wellner [38] use A +0 for B +br .) Inparticular, B br = B +br + B − br , (237)and the difference is Gaussian: B +br − B − br = Z B br ( t ) d t ∼ N (0 , / . (238)By symmetry B − br d = B +br , so we concentrate on B +br . Let ψ +br ( t ) := E e − t B +br be itsLaplace transform. Perman and Wellner [38] gave (using the notation Ψ +0 for ψ +br ) the formula Z ∞ e − xt ψ +br (cid:0) √ t / (cid:1) d t √ πt = 2 Ai( x ) x / Ai( x ) − Ai ′ ( x ) , x > , (239) . Janson/Brownian areas and Wright’s constants see also Appendix C.2.Define D + n by E ( B +br ) n = √ π − n/ n !Γ((3 n + 1) / D + n , n ≥ . (240)(Perman and Wellner [38] use L + n for our D + n and µ + n for E ( B +br ) n .) An asymp-totic expansion of the left hand side of (239) yields, arguing as for e.g. (98) and(136), 2Ai( x ) x / Ai( x ) − Ai ′ ( x ) ∼ ∞ X k =0 ( − k D + k x − k/ − / , x → ∞ , (241)and thus by (102), (103), (183), (106), (107), ∞ X k =0 D + k z k = 2 P ∞ k =0 α k z k P ∞ k =0 α k z k + P ∞ k =0 α ′ k z k = P ∞ k =0 α k z k P ∞ k =0 11 − k α k z k , (242)which leads to the recursion by Perman and Wellner [38] D + n = α n + n X k =1 k − α k D + n − k , n ≥ . (243)Using (108) and (137), we obtain from (242) also ∞ X k =0 D + k z k = 21 − P ∞ k =0 K k z k = 2 P ∞ k =0 D k z k P ∞ k =0 D k z k , (244)and the corresponding recursions D + n = n X k =1 K k D + n − k , n ≥ , (245) D + n = D n − n X k =1 D k D + n − k n ≥ . (246)Some numerical values are given in Table 6; see further Perman and Wellner[38] (but beware of typos for k = 5 and 7). We define D + ∗ n := 2 n D + n ; these areintegers by (245).Tolmatz [53] gives the asymptotics, see also Janson and Louchard [25] andcompare (156) and (155), E ( B +br ) n ∼ E B n br ∼ √ (cid:16) n e (cid:17) n/ as n → ∞ , (247)or, equivalently, D + n ∼ D n ∼ π (cid:16) (cid:17) n ( n − ∼ √ πn (cid:16) n e (cid:17) n as n → ∞ . (248) . Janson/Brownian areas and Wright’s constants E ( B +br ) = 1 E B +br = 18 q π E ( B +br ) = 120 E ( B +br ) = 714096 q π E ( B +br ) = 21118480 D +0 = 1 D +1 = 18 D +2 = 332 D +3 = 71512 D +4 = 6332048 D + ∗ = 1 D + ∗ = 1 D + ∗ = 6 D + ∗ = 71 D + ∗ = 1266 Table 6
Some numerical values for the positive part of a Brownian bridge.
Joint moments of B +br and B − br can be computed by the same method fromthe double joint Laplace transform computed by Perman and Wellner [38], see(369) in Appendix C.2 below. Taking λ = 1 in (369), the left hand side can beexpanded in an asymptotic double series, for ξ, η ց ∼ ∞ X k,l =0 ( − k + l k/ l/ √ π k ! l ! Γ (cid:16) k + l ) + 12 (cid:17) E (cid:0)(cid:0) B +br (cid:1) k (cid:0) B − br (cid:1) l (cid:1) ξ k η l . (249)Using (101) and (108), we find from (369) the identity (for formal power series),after replacing ξ by − ξ and η by − η , ∞ X k,l =0 k/ l/ √ π k ! l ! Γ (cid:16) k + l ) + 12 (cid:17) E (cid:0)(cid:0) B +br (cid:1) k (cid:0) B − br (cid:1) l (cid:1) ξ k η l = 2 (cid:18) P ∞ k =0 α ′ k ξ k P ∞ k =0 α k ξ k + P ∞ k =0 α ′ k η k P ∞ k =0 α k η k (cid:19) − = 1 − P ∞ k =0 K k ξ k − P ∞ k =0 K k η k . (250)Note that η = ξ yields (137) (using (135) and (152)) and that η = 0 yields(244).Define D ± k,l as the coefficient of ξ k η l in the left hand side of (250). Thus E (cid:0)(cid:0) B +br (cid:1) k (cid:0) B − br (cid:1) l (cid:1) = 2 − ( k + l ) / √ π k ! l !Γ (cid:0) (3( k + l ) + 1) / (cid:1) D ± k,l , k, l ≥ ∞ X k,l =0 D ± k,l ξ k η l = 1 − P ∞ k =0 K k ( ξ k + η k ) = 11 − P ∞ k =1 K k ( ξ k + η k ) , (252)which yields the recursion formula, with D ± , = 1, for k + l > D ± k,l = k X j =1 K j D ± k − j,l + l X j =1 K j D ± k,l − j . (253) . Janson/Brownian areas and Wright’s constants We have D ± k, = D ± ,k = D + k and n X k =0 D ± k,n − k = D n , n ≥ . (254)We remark further that if n = 2 m is even, then (251) and (238) yield n X j =0 ( − n − j D ± j,n − j = 2 n/ Γ (cid:0) (3 n + 1) / (cid:1) √ π n ! n X j =0 ( − n − j (cid:18) nj (cid:19) E (cid:0) B +br (cid:1) j (cid:0) B − br (cid:1) n − j = 2 n/ Γ (cid:0) (3 n + 1) / (cid:1) √ π n ! E (cid:0) B +br − B − br (cid:1) n = 2 n/ Γ (cid:0) (3 n + 1) / (cid:1) √ π n ! 12 − n/ n !2 n/ ( n/ − m − m (6 m )!(3 m )! m ! , (255)while the sum vanishes by symmetry if n is odd. Consequently, η = − ξ in (250)or (252) yields ∞ X m =0 − m − m (6 m )!(3 m )! m ! ξ m = 1 − P ∞ m =0 K m ξ m = 11 − P ∞ m =1 K m ξ m , (256)which can be written as a recursion relation for K m with even indices only.Some numerical values are given in Table 7. In particular, as found by Permanand Wellner [38], Cov( B +br , B − br ) = 1120 − π , (257)Corr( B +br , B − br ) = 128 − π − π ≈ − . . (258)We do not know of any formula for the density function f +br or distributionfunction F +br , except the Laplace inversion formulas in Tolmatz [53] and Jansonand Louchard [25] (which prove that f +br exists and is continuous). For x → ∞ we have by Tolmatz [53] and Janson and Louchard [25] asymptotic expansionsbeginning with f +br ( x ) ∼ √ √ π e − x (cid:18) x − − x − + . . . (cid:19) , (259)1 − F +br ( x ) ∼ √ π x − e − x (cid:18) − x − + 655184 x − + . . . (cid:19) . (260)We do not know any asymptotic results as x →
0, but the results on existenceof negative moments in Section 29 suggest that f +br ( x ) ≍ x − / . More preciselywe conjecture, based on (317), that f +br ( x ) ? ∼ / / Γ(1 / (2 π ) x − / , x → . (261) . Janson/Brownian areas and Wright’s constants E B , = 1 E B , = √ π E B , = 120 E B , = 71 √ π E B , = √ π E B , = 1120 E B , = 13 √ π E B , = 12880 E B , = 120 E B , = 13 √ π E B , = 43332640 E B , = 17 √ π E B , = 71 √ π E B , = 12880 E B , = 17 √ π E B , = 113564288 D ± , = 1 D ± , = 18 D ± , = 332 D ± , = 71512 D ± , = 18 D ± , = 132 D ± , = 13512 D ± , = 772048 D ± , = 332 D ± , = 13512 D ± , = 432048 D ± , = 2558192 D ± , = 71512 D ± , = 772048 D ± , = 2558192 D ± , = 302565536 Table 7
Some numerical values for the positive and negative parts of a Brownian bridge. E B k,l br is anabbreviation for E ( B +br ) k ( B − br ) l .
25. Positive part of a Brownian motion
Define B ± bm := Z B ( t ) ± d t ; (262)thus B +bm is the average of the positive part and B − bm the average of the negativepart of a Brownian motion on [0 , A + = A + (1)for B +bm .) In particular, B bm = B +bm + B − bm . (263)Note also that the difference is Gaussian: B +bm − B − bm = Z B ( t ) d t ∼ N (0 , / . (264)Again, there is symmetry: B − bm d = B +bm , and we concentrate on B +bm . Let ψ +bm ( t ) := E e − t B +bm be its Laplace transform. Perman and Wellner [38] gave (using the no-tation Ψ + for ψ +bm ) the formula Z ∞ e − xt ψ +bm (cid:0) √ t / (cid:1) d t = x − / Ai( x ) + AI( x ) x / Ai( x ) − Ai ′ ( x ) , x > , (265)see also Appendix C.1.Define L + n by E ( B +bm ) n = 2 − n/ n !Γ((3 n + 2) / L + n , n ≥ . (266) . Janson/Brownian areas and Wright’s constants E ( B +bm ) = 1 E B +bm = √ √ π E ( B +bm ) = 1796 E ( B +bm ) = 251 √ √ π E ( B +bm ) = 698940960 L +0 = 1 L +1 = 12 L +2 = 1716 L +3 = 25164 L +4 = 209671024 L + ∗ = 1 L + ∗ = 4 L + ∗ = 68 L + ∗ = 2008 L + ∗ = 83868 Table 8
Some numerical values for the positive part of a Brownian motion. (Perman and Wellner [38] use K + n for our L + n and ν + n for E ( B +bm ) n .) An asymp-totic expansion of the left hand side of (265) yields, arguing as for e.g. (98) and(182), x − Ai( x ) + x / AI( x )Ai( x ) − x − / Ai ′ ( x ) ∼ ∞ X k =0 ( − k L + k x − k/ − , x → ∞ . (267)The asymptotic expansions (102), (103), (183) of Ai( x ), Ai ′ ( x ) and AI( x ) yield,using (106) and (107), the identity ∞ X k =0 L + k z k = P ∞ k =0 α k z k + P ∞ k =0 β k z k P ∞ k =0 α k z k + P ∞ k =0 α ′ k z k = P ∞ k =0 ( α k + β k ) z k P ∞ k =0 21 − k α k z k , (268)which leads to the recursion by Perman and Wellner [38] L + n = α n + β n + n X k =1 k − α k L + n − k , n ≥ . (269)Some numerical values are given in Table 8; see further Perman and Wellner[38]. We define L + ∗ n := 2 n L + n ; these are integers as a consequence of (298)below.Janson and Louchard [25] give the asymptotics, compare (191) and (192), E ( B +bm ) n ∼ E B n bm ∼ √ (cid:16) n e (cid:17) n/ , (270) L + n ∼ L n ∼ √ (cid:16) n e (cid:17) n , as n → ∞ . (271)Joint moments of B +bm and B − bm can be computed by the same method fromthe double joint Laplace transform computed by Perman and Wellner [38], see(361) in Appendix C.1 below. Taking λ = 1 in (361), the left hand side can beexpanded in an asymptotic double series, for ξ, η ց ∼ ∞ X k,l =0 ( − k + l k/ l/ k ! l ! Γ (cid:16) k + l )2 + 1 (cid:17) E (cid:0)(cid:0) B +bm (cid:1) k (cid:0) B − bm (cid:1) l (cid:1) ξ k η l . (272) . Janson/Brownian areas and Wright’s constants Using (102), (103), (183), (106), (107), (108), (196), we find from (361) theidentity (for formal power series), after replacing ξ by − ξ and η by − η , ∞ X k,l =0 k/ l/ k ! l ! Γ (cid:16) k + l )2 + 1 (cid:17) E (cid:0)(cid:0) B +bm (cid:1) k (cid:0) B − bm (cid:1) l (cid:1) ξ k η l = (cid:18) P ∞ k =0 β k ξ k P ∞ k =0 α k ξ k + P ∞ k =0 β k η k P ∞ k =0 α k η k (cid:19) (cid:18) P ∞ k =0 α ′ k ξ k P ∞ k =0 α k ξ k + P ∞ k =0 α ′ k η k P ∞ k =0 α k η k (cid:19) − = P ∞ k =0 Q k ξ k + P ∞ k =0 Q k η k − P ∞ k =0 K k ξ k − P ∞ k =0 K k η k . (273)Note that η = ξ yields (185) and that η = 0 yields (268).Define L ± k,l as the coefficient of ξ k η l in the left hand side of (273). Thus E (cid:0)(cid:0) B +bm (cid:1) k (cid:0) B − bm (cid:1) l (cid:1) = 2 − ( k + l ) / k ! l !Γ (cid:0) k + l ) / (cid:1) L ± k,l , k, l ≥ ∞ X k,l =0 L ± k,l ξ k η l = P ∞ k =0 Q k ( ξ k + η k ) − P ∞ k =0 K k ( ξ k + η k ) , (275)which yields the recursion formula, with L ± , = 1, L ± k,l = k X j =1 K j L ± k − j,l + l X j =1 K j L ± k,l − j + δ l Q k + δ k Q l , (276)where δ is Kronecker’s delta.We have L ± k, = L ± ,k = L + k and n X k =0 L ± k,n − k = L n , n ≥ . (277)We remark further that if n = 2 m is even, then (274) and (264) yield n X j =0 ( − n − j L ± j,n − j = 2 n/ Γ(3 n/ n ! n X j =0 ( − n − j (cid:18) nj (cid:19) E (cid:0) B +bm (cid:1) j (cid:0) B − bm (cid:1) n − j = 2 n/ Γ(3 n/ n ! E (cid:0) B +bm − B − bm (cid:1) n = 2 n/ Γ(3 n/ n ! 3 − n/ n !2 n/ ( n/ m )!3 m m ! , (278)while the sum vanishes by symmetry if n is odd. Consequently, η = − ξ in (273)or (275) yields ∞ X m =0 (3 m )!3 m m ! ξ m = P ∞ m =0 Q m ξ m − P ∞ m =0 K m ξ m , (279) . Janson/Brownian areas and Wright’s constants E B , = 1 E B , = √ √ π E B , = 1796 E B , = 251 √ √ π E B , = √ √ π E B , = 196 E B , = √ √ π E B , = 149122880 E B , = 1796 E B , = √ √ π E B , = 109368640 E B , = 31 √ √ π E B , = 251 √ √ π E B , = 149122880 E B , = 31 √ √ π E B , = 9391660602880 L ± , = 1 L ± , = 12 L ± , = 1716 L ± , = 25164 L ± , = 12 L ± , = 18 L ± , = 316 L ± , = 149256 L ± , = 1716 L ± , = 316 L ± , = 109512 L ± , = 279512 L ± , = 25164 L ± , = 149256 L ± , = 279512 L ± , = 93918192 Table 9
Some numerical values for the positive and negative parts of a Brownian motion. E B k,l bm isan abbreviation for E ( B +bm ) k ( B − bm ) l . a relation between Q j and K k with even indices only.Some numerical values are given in Table 9. In particular, as found by Permanand Wellner [38], Cov( B +bm , B − bm ) = 196 − π , (280)Corr( B +bm , B − bm ) = 3 π − π − ≈ − . . (281)We do not know of any formula for the density function f +bm or distributionfunction F +bm , except the Laplace inversion formulas in Janson and Louchard[25] (which prove that f +bm exists and is continuous). For x → ∞ we have byJanson and Louchard [25] asymptotic expansions beginning with f +bm ( x ) ∼ √ √ π e − x / (cid:18) x − − x − + . . . (cid:19) , (282) P ( B +bm > x ) ∼ √ π x − e − x / (cid:18) − x − + 193648 x − + . . . (cid:19) . (283)We do not know any asymptotic results as x →
0, but the results on existenceof negative moments in Section 29 suggest that f +bm ( x ) ≍ x − / . More preciselywe conjecture, based on (319), that f +bm ( x ) ? ∼ − / / π − / Γ(1 / x − / , x → . (284) . Janson/Brownian areas and Wright’s constants
26. Convolutions and generating functions
Many of the formulas above can be written as convolutions of sequences, as isdone by Perman and Wellner [38]. This gives the simple formulas below, wherea letter X stands for the sequence ( X n ) ∞ and X ∗ Y is the sequence definedby ( X ∗ Y ) n := P nk =0 X k Y n − k . Alternatively, the formulas can be interpretedas identities for formal power series, if we instead interpret X as the generatingfunction P ∞ n =0 X n z n and ∗ as ordinary multiplication. We let denote thesequence ( δ n ) = (1 , , , . . . ) with the generating function 1.Note that the sequences denoted by various letters below, except K , all have X = 1, and thus the generating functions have constant term 1; the exceptionis K = − , which explains why − K occurs frequently. (It would be moreconsistent to give a new name to − K n and use it instead of K n in our formulas( mutatis mutandis ), but we have kept K n for historical reasons, and because allnumbers K n with n ≥ n − K n is more natural for some other purposes.)In the following list of formulas, we also give references to the correspondingequations above. Many of these formulas can also be found in Perman andWellner [38]. − α ∗ K = α ′ (108) , (70) (285) α ′ ∗ D = α (137) , (148) (286) − K ∗ D = (152) , (153) , (154) (287) α ′ ∗ L = β (185) , (186) (288) α ∗ Q = β (196) , (201) (289) − K ∗ L = Q (202) , (203) (290) D ∗ Q = L (204) , (205) (291) Q ∗ Q = W (231) , (232) (292) α ∗ α ∗ W = β ∗ β (231) (293)( α + α ′ ) ∗ D + = 2 α (242) , (243) (294) D + − K ∗ D + = 2 · (244) , (245) (295) D + + D ∗ D + = 2 D (244) , (246) (296)( α + α ′ ) ∗ L + = α + β (268) , (269) (297) D + ∗ Q + D + = 2 L + (298)The final relation (298) follows from (294), (289) and (297) by simple algebrawith generating functions.
27. Randomly stopped Brownian motion
Several of the formulas above have simple interpretations in terms of a Brownianmotion stopped at a random time, which has been found and used by Permanand Wellner [38]. . Janson/Brownian areas and Wright’s constants
Let B bm ( t ) := R t | B ( s ) | d s ; by Brownian scaling, B bm ( t ) d = t / B bm . Let fur-ther Z ∼ Exp(1) be an exponentially distributed random variable independentof the Brownian motion B . The area of the (reflected) Brownian motion stoppedat Z is thus B bm ( Z ) d = Z / B bm (299)with moments given by, see (181), E (cid:0) B bm ( Z ) (cid:1) n = E Z n/ E B n bm = 2 − n/ n ! L n . (300)Thus, L n = E (cid:0) √ B bm ( Z ) (cid:1) n n ! , n ≥ , (301)and the generating function P ∞ k =0 L k z k in (185) is the exponential generatingfunction of the moments of √ B bm ( Z ), i.e. the moment generating function of √ B bm interpreted as a formal power series. (This series diverges, so the momentgenerating function in the ordinary sense E exp (cid:0) z √ B bm ( Z ) (cid:1) is infinite for all z > √ B bm ( Z ) is given by, see (362)below, for ξ > E e − ξ √ B bm ( Z ) = Z ∞ e − t ψ bm (cid:0) √ ξt / (cid:1) d t = − AI( ξ − / ) ξ / Ai ′ ( ξ − / ) , (302)which is equivalent to (173) by a change of variables.Similarly, with obvious definitions, (266) can be written L + n = E (cid:0) √ B +bm ( Z ) (cid:1) n n ! , n ≥ , (303)and the generating function P ∞ k =0 L + k z k in (268) is the moment generating func-tion of √ B +bm interpreted as a formal power series. The Laplace transform of √ B +bm ( Z ) is given by, see (363) below, for ξ > E e − ξ √ B +bm ( Z ) = Z ∞ e − t ψ +bm (cid:0) √ ξt / (cid:1) d t = Ai( ξ − / ) + ξ − / AI( ξ − / )Ai( ξ − / ) − ξ / Ai ′ ( ξ − / ) , (304)which is equivalent to (265) by a change of variables.Furthermore, if B br ( t ), B +br ( t ) and B bm ( t ) are defined in the obvious wayusing a Brownian bridge and a Brownian meander on the interval [0 , t ], and Z ′ ∼ Γ(1 / E ( Z ′ ) n/ =Γ(3 n/ / / Γ(1 / D n = E (cid:0) √ B br ( Z ′ ) (cid:1) n n ! , n ≥ , (305) D + n = E (cid:0) √ B +br ( Z ′ ) (cid:1) n n ! , n ≥ , (306) . Janson/Brownian areas and Wright’s constants Q n = E (cid:0) √ B me ( Z ′ ) (cid:1) n n ! , n ≥ , (307)and thus P k D k z k , P k D + k z k , P k Q k z k in (137), (242), (196) are the mo-ment generating functions of √ B br ( Z ′ ), √ B +br ( Z ′ ) and √ B me ( Z ′ ), respec-tively, interpreted as (divergent) formal power series. Since Z ′ has the density π − / t − / e − t , t >
0, the Laplace transforms of these variables are given by,using (370), (371), (383) below, for ξ > E e − ξ √ B br ( Z ′ ) = Z ∞ e − t ψ br (cid:0) √ ξt / (cid:1) d t √ πt = − ξ − / Ai( ξ − / )Ai ′ ( ξ − / ) , (308) E e − ξ √ B +br ( Z ′ ) = Z ∞ e − t ψ +br (cid:0) √ ξt / (cid:1) d t √ πt = 2 Ai( ξ − / )Ai( ξ − / ) − ξ / Ai ′ ( ξ − / ) , (309) E e − ξ √ B me ( Z ′ ) = Z ∞ e − t ψ me (cid:0) √ ξt / (cid:1) d t √ πt = ξ − / AI( ξ − / )Ai( ξ − / ) , (310)which are equivalent to (133) and (134), (239) and (193).The reason for the introduction of Z ′ is that, as is well-known and explainedin greater detail by Perman and Wellner [38], if g Z := max { t ≤ Z : B ( t ) =0 } , the last zero of B before Z , then conditioned on g Z , the restrictions of B to the intervals [0 , g Z ] and [ g Z , Z ] form a Brownian bridge and a Brownianmeander on these intervals, and the lengths g Z and Z − g Z of these intervalsboth have the distribution of Z ′ . Moreover, the restrictions to these two intervalsare independent (also unconditionally, with g Z and Z − g Z independent), andthus B bm ( Z ) = B br ( g Z ) + B me ( Z − g z ), where the two variables on the righthand side are independent with the distributions of B br ( Z ′ ) and B me ( Z ′ ), whichimmediately yields (291). Considering only the positive part of B , we similarlyobtain (298) [38].For B ex , the slightly different (6) can be written K n = E (cid:0) √ B ex ( Z ′ ) (cid:1) n (6 n − n ! , n ≥ . (311)Hence, the proper analogue of L n , L + n , D n , D + n and Q n is (6 n − K n . Notethat it follows from (101) that, as x → ∞ , ∞ X k =0 ( − n (6 n − K n x − / − n/ ∼ − x (cid:18) Ai ′ ( x )Ai( x ) (cid:19) = 2 (cid:18) Ai ′ ( x )Ai( x ) (cid:19) − x, (312)the Laplace transform appearing in (84); further, in analogy with (308)–(310),(84) is equivalent to, for ξ > E e − ξ √ B ex ( Z ′ ) = Z ∞ e − t ψ ex (cid:0) √ ξt / (cid:1) d t √ πt = 2 ξ − / (cid:18) Ai ′ ( ξ − / )Ai( ξ − / ) (cid:19) − ξ − . (313) . Janson/Brownian areas and Wright’s constants For B dm we similarly let B ( t )dm be a Brownian double meander on [0 , t ]. Then B ( Z )dm has a.s. a unique zero at a time µ = τ Z where τ ∼ Beta( , ) is indepen-dent of Z ; hence µ and Z − µ = (1 − τ ) Z are two independent random variableswith a Γ(1 /
2) distribution. Moreover, see Section 23, given µ and Z − µ , B ( Z )dm consists of two independent Brownian meanders on the intervals [0 , µ ] and [ µ, Z ],with the first one reversed. Consequently, the Brownian double meander area B dm ( Z ) := R Z B ( Z )dm ( t ) d t equals the sum of two independent Brownian meanderareas, both having the same distribution as B me ( Z ′ ). Hence, by (310), E e − ξ √ B dm ( Z ) = (cid:16) E e − ξ √ B me ( Z ′ ) (cid:17) = ξ − / (cid:18) AI( ξ − / )Ai( ξ − / ) (cid:19) , (314)which is equivalent to (221) by a change of variables. Furthermore, W n = E (cid:0) √ B dm ( Z ) (cid:1) n n ! , n ≥ , (315)and the generating function P ∞ k =0 W k z k in (231) is the moment generatingfunction of √ B dm interpreted as a (divergent) formal power series.
28. A comparison
For easy reference, we collect in Table 10 the first two moments of the variousBrownian areas treated above, together with the scale invariant ratio of thesecond moment and the square of the first.Note that the variables B ex , B dm , B me , B br , B bm have quite small variances;if these variables are normalized to have means 1, their variances range from0.061 to 0.325 (in this order, for which we see no intuitive reason), which meansthat these variables, and in particular B ex , typically do not vary much fromtheir means.
29. Negative moments
We gave in Section 18 formulas for negative moments of B ex due to Flajoletand Louchard [17]. These can be generalized using the results in Appendix B.First, (339) and (340) yield by (84), (133), (173), (193), (221), (239) and (265)expressions for the negative moments of order an odd multiple of − for B ex , B br , B me , B +br , and a multiple of − for B bm , B dm , B +bm . For example, this yieldsthe values in Table 11 (including a repetition of (124) and (125)); note that(126) enables us to rewrite the values in Table 11 in other forms.More generally, by (341) and (84), (133), (173), (193), (221), where the righthand side is analytic at 0, the random variables B ex , B br , B bm , B me , B dm havenegative moments of all orders; thus, E B s ex , E B s br , E B s bm , E B s me , E B s dm areentire functions of s . . Janson/Brownian areas and Wright’s constants X E X E X E X / ( E X ) B ex √ π π ≈ . B br √ π π ≈ . B bm √ √ π
38 27 π ≈ . B me √ π π ≈ . B dm √ √ π π ≈ . B +bm √ √ π π ≈ . B +br √ π
16 120 325 π ≈ . Table 10
The first two moments for seven Brownian areas.
For B +br we see that the right hand side of (239) is finite at x = 0, but thatits derivative is ∼ cx − / as x → c = 0. Hence, (341)with m = 1, ν = 1 / x ) equal to the right hand side of (239), showsthat for 0 < s < /
2, the moment E ( B +br ) − s/ is finite if and only s > / E ( B +br ) − s is finite if and only if s < / E ( B +br ) s is an analytic function in the half-plane Re s > − . More precisely,a differentiation of (239) shows that, for this Ψ, − Ψ ′ ( x ) = √ π (cid:16) Ai(0)Ai ′ (0) (cid:17) x − / + O (1) , x → , (316)and thus the left hand side of (341) with m = 1 has the residue √ π (cid:0) Ai(0) / Ai ′ (0) (cid:1) at s = 1 /
2, and consequently (341) yieldsRes s = − E ( B +br ) s = 2 / Γ(2 / (cid:16) Ai(0)Ai ′ (0) (cid:17) = 2 / − / Γ(1 / Γ(2 / = 2 / / Γ(1 / (2 π ) . (317)For B +bm we see that the right hand side of (265) is ∼ cx − / as x → c >
0. Hence, (338) with ν = 1 and Ψ( x ) equal to the right handside of (265) shows that for 0 < s <
1, the moment E ( B +bm ) − − s ) / is finite ifand only s > /
2. Consequently, the negative moment E ( B +bm ) − s is finite if andonly if s < /
3, and E ( B +bm ) s is an analytic function in the half-plane Re s > − .(This also follows from the considerations in Section 27, which imply that, for s > E ( B +bm ( Z )) − s is finite if and only if E ( B +br ( Z ′ )) − s = E ( B +br ) − s E ( Z ′ ) − s/ is, and E ( Z ′ ) − s/ < ∞ if and only if s < / . Janson/Brownian areas and Wright’s constants E B − / = 2 / √ π Γ(1 / · (cid:16) Ai ′ (0)Ai(0) (cid:17) = 2 − / / π − / Γ(2 / ≈ E B − / = 2 − / √ π Γ(1 / · Ai(0) − Ai ′ (0) = 2 − / / π − / Γ(1 / ≈ E B − / = 2 − / √ π Γ(1 / · AI(0)Ai(0) = 2 − / / π − / Γ(2 / ≈ E ( B +br ) − / = 2 / √ π Γ(1 / · Ai(0) − Ai ′ (0) = 2 − / / π − / Γ(1 / ≈ E B − / = 2 − / / · AI(0) − Ai ′ (0) = 2 − / / Γ(1 / / ≈ E B − / = 2 − / / · (cid:16) AI(0)Ai(0) (cid:17) = 2 − / / Γ(2 / ≈ E ( B +bm ) − / = ∞ E B − = 3 √ π (cid:18) − / Γ(2 / π (cid:19) ≈ E B − = 2 − / √ π ≈ E B − = 2 − / √ π (cid:18) − / Γ(2 / π (cid:19) ≈ E ( B +br ) − = ∞ E B − / = 2 − / / π − Γ(1 / ≈ E B − / = 2 − / / π − Γ(2 / (cid:16) − / Γ(2 / π (cid:17) ≈ E B − = 3 + 3 / Γ(1 / π ≈ E B − = 6 − / Γ(2 / π + 27 Γ(2 / π ≈ Table 11
Some negative fractional moments.. Janson/Brownian areas and Wright’s constants that, for this Ψ, Ψ( x ) = − Ai(0)Ai ′ (0) x − / + O (1) , x → , (318)and thus the left hand side of (338) has the residue − Ai(0) / Ai ′ (0) at s = 1 / s = − E ( B +bm ) s = 2 / Γ(1 / /
3) Ai(0) − Ai ′ (0) = 2 / − / √ π Γ(2 / − / / π − / Γ(1 / . (319)Furthermore, the right hand sides of (239) and (265) have expansions inpowers of x / at x = 0 (beginning with x − / for (265)). Hence, in (341) withΨ( x ) equal to one of these functions, the left hand side extends to a meromorphicfunction in 0 < Re s < m + ν with possible poles (all simple) only at , , . . . , andthus the same holds for the negative moment E X − m + ν − s ) / on the right handside of (341). Since m is an arbitrary non-negative integer and ν = 1 / B +br while ν = 1 for B +bm , it follows that E ( B +br ) s extends to a meromorphic functionin the complex plane with poles only at − , − , −
2, . . . , i.e., multiples of − , and that E ( B +bm ) s extends to a meromorphic function in the complex planewith poles only at − , − − , . . . , i.e., odd multiples of − . Furthermore,all poles are simple. (We cannot prove that all these points really are poles;it is conceivable that some actually are regular points because some terms inthe expansions of Ψ( x ) might vanish due to unexpexted cancellations, but webelieve that all actually are poles. At least, − and − are poles by the resultsabove.)Recall also the relation (122) in Section 18 between moments of B ex andthe root zeta function Λ of Ai. For the Brownian bridge, there is an analogousrelation between moments of B br and the root zeta function of Ai ′ defined by,cf. (120), ˜Λ( s ) := ∞ X j =1 | a ′ j | − s , Re s > / . (320)Since | a ′ j | ∼ | a j | ∼ (3 πj/ / [1, 10.4.95] (in fact, | a j − | < | a ′ j | < | a j | for j ≥ s ) is analytic for Re s > /
2. By (335) in Appendix B . Janson/Brownian areas and Wright’s constants and (159), for Re s > s ) E B − s br = Z ∞ t s − ψ br ( t ) d t = 2 − / √ π Z ∞ t s +1 / ∞ X j =1 | a ′ j | − exp (cid:0) − − / | a ′ j | t / (cid:1) d tt = 2 − / √ π ∞ X j =1 | a ′ j | − Z ∞ u (3 s +1) / exp (cid:0) − − / | a ′ j | u (cid:1) d uu = 2 s/ − √ π ∞ X j =1 | a ′ j | − (3 s +3) / Γ (cid:16) s + 12 (cid:17) = 3 √ π s/ − Γ (cid:16) s + 12 (cid:17) ˜Λ (cid:16) s + 32 (cid:17) . (321)Since E B s br is an entire function, this shows that ˜Λ extends to a meromorphicfunction in the complex plane with, similarly to (122), E B s br = 3 √ π − s/ − Γ( − s ) Γ (cid:16) − s (cid:17) ˜Λ (cid:16) − s (cid:17) , s ∈ C . (322)Since the left hand side has no poles and no real zeros, it follows, arguing as inSection 18, that ˜Λ( z ) has simple poles at z = , − , − , . . . , and zeros at 1, − − − − −
7, . . . , and nowhere else.It follows from (158) that, for | z | < | a ′ | ,Ai( z )Ai ′ ( z ) = X m ≥ ( − m +1 ˜Λ( m + 2) z m , (323)and thus the values of ˜Λ( s ) at positive integers s = 2 , , . . . can be com-puted from the Taylor series of Ai. (In particular, ˜Λ(2) = − Ai(0) / Ai ′ (0) =3 − / Γ(1 / / Γ(2 /
3) and ˜Λ(3) = 1. Furthermore, as just said, ˜Λ(1) = 0.) Thisand (322) give again the same explicit formulas that we have obtained from(340) for the negative moments B − s br when s is an odd multiple of 1 / < Re s < / Z ∞ x s − Ai( x )Ai ′ ( x ) d x = − / s/ √ π Γ( s )Γ (cid:16) − s (cid:17) E B (2 s − / = − Γ( s )Γ(1 − s )˜Λ(2 − s )= − π sin( πs ) ˜Λ(2 − s ) , (324)which also follows from (158); cf. (127) and (128).For Brownian motion, Brownian meander and Brownian double meander,more complicated analogues of the root zeta function Λ( s ) appear in the same . Janson/Brownian areas and Wright’s constants way: for Brownian motion, using (170), P ∞ j =1 κ j | a ′ j | − s ; for Brownian mean-der, using (209), P ∞ j =1 r j | a j | − s ; for Brownian double meander, using (220), P ∞ j =1 r j | a j | − s . We do not pursue this further. Appendix A: The integrated Airy function
We define, as in (78), using R ∞ Ai( x ) d x = 1 / z ) := Z + ∞ z Ai( t ) d t = 13 − Z z Ai( t ) d t ; (325)this is well-defined for all complex z and yields an entire function provided thefirst integral is taken along, for example, a path that eventually follows thepositive real axis to + ∞ . Note that AI ′ ( z ) = − Ai( z ) and that AI(0) = 1 / x → + ∞ AI( x ) = 0 , (326)lim x →−∞ AI( x ) = Z ∞−∞ Ai( x ) d x = 1 . (327)In terms of the functions Gi and Hi defined in [1], we have, see [1, 10.4.47–48],AI( z ) = π (cid:0) Ai( z )Gi ′ ( z ) − Ai ′ ( z )Gi( z ) (cid:1) (328)= 1 + π (cid:0) Ai ′ ( z )Hi( z ) − Ai( z )Hi ′ ( z ) (cid:1) . (329)Repeated integrations by parts giveAI( z ) = Z + ∞ z Ai( w ) d w = Z + ∞ z w Ai ′′ ( w ) d w = − Ai ′ ( z ) z + Z + ∞ z w Ai ′ ( w ) d w = − Ai ′ ( z ) z − Ai( z ) z + Z + ∞ z w Ai( w ) d w = − Ai ′ ( z ) z − Ai( z ) z + Z + ∞ z w Ai ′′ ( w ) d w = . . . ; (330)if we assume | arg( z ) | < π − δ for some δ >
0, we can here integrate for examplealong the horizontal line { z + t : t ≥ } , and it follows from the asymptoticexpansions (102) and (103), which are valid for | arg( z ) | < π − δ [1, 10.4.59 and10.4.61] that there is a similar asymptotic expansion, as | z | → ∞ ,AI( z ) ∼ √ π z − / e − z / / ∞ X k =0 ( − k β k z − k/ , (331)valid for | arg( z ) | < π − δ and thus extending (183) (which is given in Tak´acs[49]; see also [1, 10.4.82]). The coefficients β k can be found by this procedure, . Janson/Brownian areas and Wright’s constants but it is easier to observe that we can formally differentiate (331) and obtainan asymptotic expansion of − Ai( z ), and a comparison with (102) yields therecursion relation (184). See Table 3 for a few numerical values.Similarly, along the negative real axis, and more generally for − z where | arg( z ) | < π/ − δ for some δ >
0, there are asymptotic expansions [1, 10.4.60and 10.4.62]Ai( − z ) ∼ π − / (cid:18) sin (cid:16) z / + π (cid:17) ∞ X k =0 ( − k α k z − k − / − cos (cid:16) z / + π (cid:17) ∞ X k =0 ( − k α k +1 z − k − / (cid:19) , (332)Ai ′ ( − z ) ∼ − π − / (cid:18) cos (cid:16) z / + π (cid:17) ∞ X k =0 ( − k α ′ k z − k +1 / + sin (cid:16) z / + π (cid:17) ∞ X k =0 ( − k α ′ k +1 z − k − / (cid:19) , (333)which using (330) lead to, as | z | → ∞ with | arg( z ) | < π/ − δ ,AI( − z ) ∼ − π − / (cid:18) cos (cid:16) z / + π (cid:17) ∞ X k =0 ( − k β k z − k − / + sin (cid:16) z / + π (cid:17) ∞ X k =0 ( − k β k +1 z − k − / (cid:19) ; (334)given (at least along the negative real axis) by Tak´acs [49, (17)] (using his h k = (2 / k β k ); see also [1, 10.4.83]. Appendix B: The Mellin transform of a Laplace transform
For any positive random variable X with Laplace transform ψ ( t ) := E e − tX ,and any s >
0, Fubini’s theorem yields Z ∞ t s − ψ ( t ) d t = E Z ∞ t s − e − tX d t = Γ( s ) E X − s . (335)If one side is finite for s = s >
0, and thus both sides are, then E X − s < ∞ for s = s and thus for 0 ≤ s ≤ s ; hence both sides of (335) are finite and(335) holds for all s with 0 < s ≤ s , and more generally for all complex s with0 < Re( s ) ≤ s , with the terms in (335) analytic in the strip 0 < Re( s ) < s . Lemma B.1.
The following are equivalent: (i) X has negative moments of all orders, i.e., E X − s < ∞ for all s > ; . Janson/Brownian areas and Wright’s constants (ii) R ∞ t s − ψ ( t ) d t < ∞ for all s > ; (iii) ψ ( t ) = O ( t − N ) as t → ∞ for all N .If these hold and further X has moments of all (positive) orders, then E X − s is an entire function of s ; further, (335) holds for Re s > and the right handside defines a meromorphic extension of R ∞ t s − ψ ( t ) d t to the complex plane.Proof. (i) ⇐⇒ (ii) by (335).Since ψ ( t ) ≤ ψ (0) = 1, (iii) = ⇒ (ii).Conversely, since ψ is decreasing, (ii) implies for every u > ∞ > Z ∞ t s − ψ ( t ) d t ≥ ψ ( u ) Z u t s − d t = s − u s ψ ( u ) , (336)and thus ψ ( u ) = O ( u − s ).The last statement is obvious.Several formulas above, viz. (84), (133), (173), (193), (221), (239), (265), givea formula for a double Laplace transform of the form Z ∞ e − xt ψ (cid:0) √ t / (cid:1) t ν − d t = Ψ( x ) , x > , (337)where ν = 1 or 1 /
2; see also Section 27 for an explanation of this form. (Wehave ν = 1 for B bm and B +bm and ν = 1 / < s < ν , and if theresult then is finite also for complex s with 0 < Re s < ν , Z ∞ x s − Ψ( x ) d x = Γ( s ) Z ∞ t − s ψ (cid:0) √ t / (cid:1) t ν − d t = Γ( s ) 23 Z ∞ − ( ν − s ) / u ν − s ) / − ψ ( u ) d u = 2 − ( ν − s ) / s )Γ (cid:16) ν − s )3 (cid:17) E X − ν − s ) / . (338)Furthermore, x = 0 in (337) similarly yields, with Ψ(0) := lim x → Ψ( x ) ≤ ∞ ,Ψ(0) = Z ∞ ψ (cid:0) √ t / (cid:1) t ν − d t = 23 Z ∞ − ν/ u ν/ − ψ ( u ) d u = 2 − ν/ (cid:16) ν (cid:17) E X − ν/ , (339)which can be regarded as a limiting case of (338) for s →
0. In particular,Ψ(0) < ∞ if and only if E X − ν/ < ∞ (which holds for all Brownian areastreated here except B +bm ). . Janson/Brownian areas and Wright’s constants More generally, by first taking the m :th derivative of (337), we obtain forevery integer m ≥
0, by letting x → ( m ) (0) := lim x → Ψ ( m ) ( x ),( − m Ψ ( m ) (0) = Z ∞ ψ (cid:0) √ t / (cid:1) t m + ν − d t = 2 − ( m + ν ) / (cid:16) m + ν )3 (cid:17) E X − m/ − ν/ (340)and, for 0 < s < m + ν (and for 0 < Re s < m + ν if the result is finite),( − m Z ∞ x s − Ψ ( m ) ( x ) d x = 2 − ( m + ν − s ) / s )Γ (cid:16) m + ν − s )3 (cid:17) E X − m + ν − s ) / . (341) Appendix C: Feynman-Kac formulas
Many of the results above are derived from formulas obtained by simple ap-plications of the Feynman–Kac formula, following Kac [30; 31]. This is by nowstandard, but for completeness we give here a unified treatment of all the Brow-nian functionals discussed above. We basically follow the proof by Shepp [42] forthe Brownian bridge. See also Jeanblanc, Pitman and Yor [27] for a treatmentof several related Brownian functionals.One version of the Feynman–Kac formula is the following. Let B x ( t ) denotethe Brownian motion started at x ; thus B x ( t ) = x + B ( t ); we continue to use B ( t ) for B ( t ). Lemma C.1.
Let V ( x ) ≥ be a non-negative continuous function on ( −∞ , ∞ ) ,let λ > , and let φ + and φ − be C solutions of the differential equation φ ′′ ( x ) = (cid:0) V ( x ) + λ (cid:1) φ ( x ) (342) such that φ + is bounded on [0 , ∞ ) and φ − is bounded on ( −∞ , . Let w := φ + (0) φ ′− (0) − φ − (0) φ ′ + (0) and assume that w = 0 . Moreover, let f be anybounded function on R . Then, for all real x , Z ∞ e − λt E (cid:16) e − R t V ( B x ( s )) d s f (cid:0) B x ( t ) (cid:1)(cid:17) d t = 2 w (cid:18) φ + ( x ) Z x −∞ φ − ( y ) f ( y ) d y + φ − ( x ) Z ∞ x φ + ( y ) f ( y ) d y (cid:19) . (343) Proof.
Although this is well-known, we give a proof for completeness.Note that if we define the Wronskian W ( x ) := φ + ( x ) φ ′− ( x ) − φ − ( x ) φ ′ + ( x ) , (344)then (342) implies that W ′ ( x ) = 0 and thus W ( x ) is constant; hence W ( x ) = w. (345) . Janson/Brownian areas and Wright’s constants Assume first that f is continuous and has compact support, and define φ ( x ) := φ + ( x ) Z x −∞ φ − ( y ) f ( y ) d y + φ − ( x ) Z ∞ x φ + ( y ) f ( y ) d y. (346)Then φ ∈ C ( R ), and differentiation yields φ ′ ( x ) = φ ′ + ( x ) Z x −∞ φ − ( y ) f ( y ) d y + φ ′− ( x ) Z ∞ x φ + ( y ) f ( y ) d y. (347)Hence also φ ′ ∈ C , and thus φ ∈ C . A second differentiation yields, using(342), φ ′′ ( x ) = 2 (cid:0) V ( x ) + λ (cid:1) φ ( x ) − W ( x ) f ( x ) = 2 (cid:0) V ( x ) + λ (cid:1) φ ( x ) − wf ( x ) . (348)Moreover, since f has compact support, (346) shows that if x is large enough,then φ ( x ) = (cid:0)R φ − f (cid:1) φ + ( x ), and thus φ ( x ) is bounded for large x ; similarly φ ( x )is bounded for x → −∞ . Consequently, φ is bounded on R .Let X ( t ) and Y ( t ) be the stochastic processes X ( t ) := λt + R t V (cid:0) B x ( s ) (cid:1) d s and Y ( t ) := e − X ( t ) φ (cid:0) B x ( t ) (cid:1) + w Z t e − X ( u ) f (cid:0) B x ( u ) (cid:1) d u. (349)A straightforward application of Itˆo’s formula, using (348), shows thatd Y ( t ) = e − X ( t ) φ ′ (cid:0) B x ( t ) (cid:1) d B x ( t ) , (350)and thus Y ( t ) is a local martingale. Moreover, since X t ≥ λt and φ and f arebounded, Y ( t ) is uniformly bounded, and thus a bounded martingale. Hence, φ ( x ) = Y (0) = E Y ( ∞ ) , (351)where, recalling that φ is bounded and X ( t ) ≥ λt , Y ( ∞ ) := lim t →∞ Y ( t ) = w Z ∞ e − X ( u ) f (cid:0) B x ( u ) (cid:1) d u. (352)The result (343) now follows by (351), (352), and (346), under our assumptionthat f is continuous with compact support.By a monotone class theorem, for example [20, Theorem A.1], (343) remainstrue for all bounded measurable f with support in a finite interval [ − A, A ], forany
A < ∞ . Fixing an x with φ − ( x ) = 0 and letting f ( y ) := sign( φ + ( y )) [ x Majumdar and Comtet [35] have given “physical proofs” of (80)and (209) using path integral techniques. This method is closely related tothe Feynman–Kac method as formulated in Lemma C.1, which can be seen as . Janson/Brownian areas and Wright’s constants follows, where we argue formally and ignore giving proper technical conditionson V and f for the validity of the argument.Let H be the differential operator Hφ := − φ ′′ + V φ . (This is an unboundedpositive self-adjoint operator in L ( R ).) Then (348) can be written Hφ + λφ = w f. (353)Hence, the right hand side of (343), which is w φ ( x ) by (346), equals ( H + λ ) − f ( x ). Further, ( H + λ ) − f = R ∞ e − λt e − tH f d t , so by the uniqueness theo-rem for Laplace transforms, (343) is equivalent to E (cid:16) e − R t V ( B x ( s )) d s f (cid:0) B x ( t ) (cid:1)(cid:17) = e − tH f ( x ) , (354)which essentially is the path integral formula used by Majumdar and Comtet[35] in their proof.We let ξ, η > V ( x ) := ( √ ξx, x ≥ , √ η | x | , x < . (355)It is easily checked that Ai (cid:0) √ ξ / x + ξ − / λ (cid:1) solves (342) on (0 , ∞ ) and isbounded there; thus this can be extended to a solution φ + on R that is boundedon [0 , ∞ ). Similarly, Ai (cid:0) −√ η / x + η − / λ (cid:1) on ( −∞ , 0) can be extended toa solution φ − that is bounded on ( −∞ , φ ± as in Lemma C.1with φ + ( x ) = Ai (cid:0) √ ξ / x + ξ − / λ (cid:1) , x ≥ , (356) φ − ( x ) = Ai (cid:0) √ η / | x | + η − / λ (cid:1) , x ≤ . (357)As a consequence, w = −√ η / Ai ′ ( η − / λ )Ai( ξ − / λ ) − √ ξ / Ai ′ ( ξ − / λ )Ai( η − / λ ) . (358)We will now apply Lemma C.1 with this V to the different cases. C.1. Brownian motion Recall the definition (262) of B ± bm and define the joint Laplace transform, for ξ, η ≥ ψ ± bm ( ξ, η ) := E e − ξ B +bm − η B − bm . (359)Thus ψ bm ( ξ ) = ψ ± bm ( ξ, ξ ) and ψ +bm ( ξ ) = ψ ± bm ( ξ, ξ, η > V as in (355), φ ± as in (356)and (357), f = 1 and x = 0. By (355) and homogeneity, Z t V ( B ( s )) d s d = t / Z V ( B ( s )) d s = √ t / ξ B +bm + √ t / η B − bm , (360) . Janson/Brownian areas and Wright’s constants and thus (343) yields, recalling (359) and using (356), (357), (358) and (78), Z ∞ e − λt ψ ± bm (cid:0) √ t / ξ, √ t / η (cid:1) d t = 2 w (cid:18) φ + (0) Z −∞ φ − ( y ) d y + φ − (0) Z ∞ φ + ( y ) d y (cid:19) = 2 w (cid:18) Ai( ξ − / λ ) AI( η − / λ ) √ η / + Ai( η − / λ ) AI( ξ − / λ ) √ ξ / (cid:19) = (cid:18) AI( ξ − / λ ) ξ / Ai( ξ − / λ ) + AI( η − / λ ) η / Ai( η − / λ ) (cid:19) (cid:18) − ξ / Ai ′ ( ξ − / λ )Ai( ξ − / λ ) − η / Ai ′ ( η − / λ )Ai( η − / λ ) (cid:19) − . (361)In particular, the special case η = ξ yields Z ∞ e − λt ψ bm (cid:0) √ t / ξ (cid:1) d t = − AI( ξ − / λ ) ξ / Ai ′ ( ξ − / λ ) . (362)Similarly, letting η → x ) / Ai( x ) ∼ x − / and Ai ′ ( x ) / Ai( x ) ∼− x / as x → ∞ by (102), (103) and (183), Z ∞ e − λt ψ +bm (cid:0) √ t / ξ (cid:1) d t = (cid:18) AI( ξ − / λ ) ξ / Ai( ξ − / λ ) + λ − / (cid:19) (cid:18) − ξ / Ai ′ ( ξ − / λ )Ai( ξ − / λ ) + λ / (cid:19) − = ξ − / AI( ξ − / λ ) + λ − / ξ / Ai( ξ − / λ ) − ξ / Ai ′ ( ξ − / λ ) + λ / Ai( ξ − / λ )= λ − Ai( ξ − / λ ) + ξ − / λ / AI( ξ − / λ )Ai( ξ − / λ ) − ξ / λ − / Ai ′ ( ξ − / λ ) . (363)Taking ξ = 1 yields (265). C.2. Brownian bridge Let B ( t ) x,y denote the Brownian bridge on [0 , t ] with boundary values B ( t ) x,y (0) = x and B ( t ) x,y ( t ) = y ; it can be defined by conditioning B x on { B x ( t ) = y } ,interpreted in the usual way (with a distribution that is a continuous functionof y ). The standard Brownian bridge B br equals B (1)0 , .The left hand side of (343) can be written, by conditioning on B x ( t ) insidethe integral, Z ∞ t =0 e − λt Z ∞ y = −∞ E e − R t V ( B ( t ) x,y ( s )) d s f ( y ) e − ( y − x ) / (2 t ) d y √ πt d t (364) . Janson/Brownian areas and Wright’s constants while the right hand side can be written, with x ∨ y := max( x, y ) and x ∧ y :=min( x, y ), 2 w Z ∞−∞ φ + ( x ∨ y ) φ − ( x ∧ y ) f ( y ) d y. (365)Hence, using also Fubini in (364), Z ∞ e − λt E e − R t V ( B ( t ) x,y ( s )) d s e − ( y − x ) / (2 t ) d t √ πt = 2 w φ + ( x ∨ y ) φ − ( x ∧ y ) (366)for a.e. y . However, both sides of (366) are continuous functions of y (the lefthand side by dominated convergence because B ( t ) x,y ( u ) d = B ( t ) x, ( u )+ yu/t in C [0 , t ]and thus B ( t ) x,y n d −→ B ( t ) x,y in C [0 , t ] as y n → y ). Hence, (366) holds for every y ∈ R .Take x = y = 0 and V as in (355). Then, by homogeneity, recalling (236), Z t V (cid:0) B ( t )0 , ( s ) (cid:1) d s d = t / Z V (cid:0) B br ( s ) (cid:1) d s = √ t / ξ B +br + √ t / η B − br . (367)Define the joint Laplace transform, ψ ± br ( ξ, η ) := E e − ξ B +br − η B − br ; (368)thus ψ br ( ξ ) = ψ ± br ( ξ, ξ ) and ψ +br ( ξ ) = ψ ± br ( ξ, ξ, η ≥ Z ∞ e − λt ψ ± br (cid:0) √ t / ξ, √ t / η (cid:1) d t √ πt = 2 w φ + (0) φ − (0)= √ (cid:18) − ξ / Ai ′ ( ξ − / λ )Ai( ξ − / λ ) − η / Ai ′ ( η − / λ )Ai( η − / λ ) (cid:19) − . (369)In particular, taking ξ = η > Z ∞ e − λt ψ br (cid:0) √ t / ξ (cid:1) d t √ πt = − Ai( ξ − / λ ) ξ / Ai ′ ( ξ − / λ ) , (370)which by simple changes of variables is equivalent to (133) ( λ = x , ξ = 1) and(134) ( λ = x , ξ = u/ √ η → Z ∞ e − λt ψ +br (cid:0) √ t / ξ (cid:1) d t √ πt = 2 λ − / Ai( ξ − / λ )Ai( ξ − / λ ) − ξ / λ − / Ai ′ ( ξ − / λ ) . (371)Taking ξ = 1 yields (239). . Janson/Brownian areas and Wright’s constants C.3. Brownian meander We apply again Lemma C.1 with V given by (355); this time keeping ξ > η → ∞ ; we also assume x > 0. By monotone convergence, theleft hand side of (343) tends to Z ∞ e − λt E (cid:16) e −√ ξ R t B x ( s ) d s f (cid:0) B x ( t ) (cid:1) ; B x ( s ) ≥ , t ] (cid:17) d t = Z ∞ e − λt E (cid:16) e −√ ξ R t B x ( s ) d s f (cid:0) B x ( t ) (cid:1) (cid:12)(cid:12) B x ( s ) ≥ , t ] (cid:17) P x ( τ > t ) d t, (372)where P x ( τ > t ) is the probability that the hitting time τ := inf { t > B x ( t ) = 0 } is larger than t , for the Brownian motion B x started at x . For theright hand side of (343), let us assume for simplicity that f ( y ) = 0 for all y < x .We note that by (356) and (357), φ + ( y ) does not depend on η for y ≥ 0, while φ − ( y ) does. On the half-line [0 , ∞ ), φ − is by (342) governed by the equation φ ′′ ( x ) = (cid:0) √ ξx + λ (cid:1) φ ( x ); let φ and φ be the solutions to this equation with φ (0) = 1 , φ ′ (0) = 0 , (373) φ (0) = 0 , φ ′ (0) = 1 . (374)(Note that these do not depend on η .) Thus, for x > φ − ( x ) = φ − (0) φ ( x ) + φ ′− (0) φ ( x ) . (375)Furthermore, as η → ∞ , (357) implies φ − (0) → Ai(0) , φ ′− (0) ∼ −√ η / Ai ′ (0) , (376)while, by (358), w ∼ −√ η / Ai ′ (0)Ai( ξ − / λ ) . (377)As η → ∞ , we thus have by (375), (376), (377), φ − ( x ) w → φ ( x )Ai( ξ − / λ ) = φ ( x ) φ + (0) . (378)Returning to (343) we thus have, for x > f ( y ) = 0 for y < x ,see also [19, Corollary 2.1(ii)], Z ∞ e − λt E (cid:16) e −√ ξ R t B x ( s ) d s f (cid:0) B x ( t ) (cid:1) (cid:12)(cid:12) B x ( s ) ≥ , t ] (cid:17) P x ( τ > t ) d t = 2 φ ( x ) φ + (0) Z ∞ x φ + ( y ) f ( y ) d y. (379)Now divide by x and let x ց 0. The distribution of B x conditioned on B x ( s ) ≥ , t ] converges to the distribution of the Brownian meander B ( t )me . Janson/Brownian areas and Wright’s constants on [0 , t ]. Furthermore, by the well-known formula for the distribution of thehitting time, with Φ the normal distribution function,1 x P x ( τ > t ) = 1 x (cid:16) (cid:16) x √ t (cid:17) − (cid:17) → √ πt , (380)which also gives the bound1 x P x ( τ > t ) = 1 x (cid:16) (cid:16) x √ t (cid:17) − (cid:17) ≤ √ πt , (381)while the expectation inside the integral on the left hand side of (379) is boundedby sup | f | , so dominated convergence applies to the left hand side of (379). Forthe right hand side we have φ ( x ) /x → φ ′ (0) = 1. Consequently, (379) yields Z ∞ e − λt E e −√ ξ R t B ( t )me ( s ) d s f (cid:0) B ( t )me ( t ) (cid:1) √ πt d t = 2 φ + (0) Z ∞ φ + ( y ) f ( y ) d y (382)for every f that is bounded and continuous on (0 , ∞ ) and vanishes on someinterval (0 , δ ), and thus by a limit argument (for example a monotone classargument) for every bounded f on (0 , ∞ ). Taking f ( x ) = 1 / √ B ( t )me ( ts ) d = t / B me ( s ), 0 ≤ s ≤ Z ∞ e − λt ψ me (cid:0) √ ξt / (cid:1) d t √ πt = √ φ + (0) Z ∞ φ + ( y ) d y = ξ − / AI( ξ − / λ )Ai( ξ − / λ ) . (383)Taking ξ = 1 yields (193) and λ = 1 yields (310).Note also that, since B ( t )me ( t ) has the density function ( y/t ) e − y / (2 t ) , y > 0, itfollows from (382) that for every y > Z ∞ e − λt E (cid:16) e −√ ξ R t B ( t )me ( s ) d s (cid:12)(cid:12)(cid:12) B ( t )me ( t ) = y (cid:17) yt e − y / (2 t ) √ πt d t = φ + ( y ) φ + (0) = Ai (cid:0) √ ξ / y + ξ − / λ (cid:1) Ai (cid:0) ξ − / λ (cid:1) , (384)which is a relation for the areas under the Bessel bridges (cid:0) B ( t )me | B ( t )me ( t ) = y (cid:1) . C.4. Brownian excursion Formula (384) holds for every λ > 0, and by monotone convergence as λ → λ = 0 too. Taking the difference, we obtain . Janson/Brownian areas and Wright’s constants Z ∞ (cid:0) − e − λt (cid:1) E (cid:16) e −√ ξ R t B ( t )me ( s ) d s (cid:12)(cid:12)(cid:12) B ( t )me ( t ) = y (cid:17) yt e − y / (2 t ) √ πt d t = Ai (cid:0) √ ξ / y (cid:1) Ai(0) − Ai (cid:0) √ ξ / y + ξ − / λ (cid:1) Ai (cid:0) ξ − / λ (cid:1) . (385)Let ξ = 1 / √ 2, divide by y and let y → 0, noting that as y → (cid:0) B ( t )me (cid:12)(cid:12) B ( t )me ( t ) = y (cid:1) d −→ B ( t )ex , a Brownian excursion on [0 , t ]. We then find by dominatedconvergence on the left hand side, and simple differentiation with respect to y on the right hand side, the formula (81), which as explained in Section 13 yields(83) and (84).Alternatively, we may take ξ = 1 in (384), differentiate with respect to λ andthen divide by y and let y → 0. 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