WWINDINGS OF THE STABLE KOLMOGOROV PROCESS
CHRISTOPHE PROFETA AND THOMAS SIMON
Abstract.
We investigate the windings around the origin of the two-dimensional Markovprocess (
X, L ) having the stable L´evy process L and its primitive X as coordinates, in thenon-trivial case when | L | is not a subordinator. First, we show that these windings have analmost sure limit velocity, extending McKean’s result [8] in the Brownian case. Second, weevaluate precisely the upper tails of the distribution of the half-winding times, connectingthe results of our recent papers [9, 10]. Introduction and statement of the results
A celebrated theorem by F. Spitzer [12] states that the angular part { ω ( t ) , t ≥ } of atwo-dimensional Brownian motion starting away from the origin satisfies the following limittheorem 2 ω ( t )log t d −→ C as t → + ∞ , where C denotes the standard Cauchy law. An analogue of this result for isotropic stableL´evy processes was given in [2], with a slower speed in √ log t and a centered Gaussian limitlaw. Notice that both these results can be obtained as functional limit theorems with respectto the Skorohod topology. We refer to [4] for a recent paper revisiting these problems, withfurther results and an updated bibliography.In a different direction, McKean [8] had observed that the windings of the Kolmogorovdiffusion, which is the two-dimensional process Z having a linear Brownian motion as secondcoordinate and its running integral as first coordinate, obey an almost sure limit theorem.More precisely, if { ω ( t ) , t ≥ } denotes the angular part of the process Z starting away fromthe origin, it is shown in Section 4.3 of [8] that ω ( t )log t a.s. −→ − √
32 as t → + ∞ (the constant which is given in [8] is actually −√ / , but it will be observed below that theevaluation of the relevant improper integral in [8] was slightly erroneous). Of course, thedegeneracy of the Kolmogorov diffusion makes it wind in a very particular way, since thisprocess visits a.s. alternatively and clockwise the left and right half-planes. The regularityof this behaviour, which contrasts sharply with the complexity of planar Brownian motion, Mathematics Subject Classification.
Key words and phrases.
Integrated process - Harmonic measure - Hitting time - Stable L´evy process -Winding. a r X i v : . [ m a t h . P R ] J u l CHRISTOPHE PROFETA AND THOMAS SIMON makes it possible to use the law of large numbers and to get an almost sure limit theorem.The first aim of this paper is to obtain an analogue of McKean’s result in replacingBrownian motion by a strictly α − stable L´evy process L = { L t , t ≥ } . Without loss ofgenerality, we choose the following normalization for the characteristic exponentΨ( λ ) = log( E [ e i λL ]) = − (i λ ) α e − i παρ sgn( λ ) , λ ∈ R , (1.1)where α ∈ (0 ,
2] is the self-similarity parameter and ρ = P [ L ≥
0] is the positivity parameter.We refer to [11, 13] for accounts on stable laws and processes, and to the introduction of ourprevious paper [10] for a discussion on this specific parametrization. Recall that if α = 2 , then necessarily ρ = 1 / L = {√ B t , t ≥ } is a rescaled Brownian motion. Introducethe primitive process X t = (cid:90) t L s ds, t ≥ , and denote by P ( x,y ) the law of the strong Markov process Z = ( X, L ) started from ( x, y ). Byanalogy with the classical Kolmogorov diffusion [6], this process may and will be called thestable Kolmogorov process. When ( x, y ) (cid:54) = (0 , , it can be shown without much difficulty- see Lemma 3 below - that under P ( x,y ) , the process Z never hits (0 , . Filling in the gapsmade by the jumps of L by vertical lines - see the figure below - and reasoning exactly as in[2] p.1270 it is possible to define the algebraic angle ω ( t ) = (cid:92) ( Z , Z t )measured in the trigonometric orientation. Figure 1.
One path of (( X t , L t ) , t ≤ X = − L = 0. INDINGS OF THE STABLE KOLMOGOROV PROCESS 3 If ρ = 1 resp. ρ = 0, then | L | is a stable subordinator and it is easy to see that Z stays forlarge times within the positive resp. the negative quadrant with a.s. X t /L t → + ∞ , so that ω ( t ) converges a.s. to a finite limit which is (cid:92) ( Z , O x ) resp. (cid:92) ( − Z , O x ) . When ρ ∈ (0 ,
1) and ( x, y ) (cid:54) = (0 , , the L´evy process L oscillates and the Kolmogorov process Z winds clockwise and infinitely often around the origin as soon as ( x, y ) (cid:54) = (0 , . Indeed,considering the partition R \{ (0 , } = P − ∪ P + with P − = { x < } ∪ { x = 0 , y < } and P + = { x > } ∪ { x = 0 , y > } , we see that if ( x, y ) ∈ P − the continuous process X visits alternatively the negative andpositive half-lines, starting negative, and that its speed when it hits zero is alternativelypositive and negative, starting positive. When ( x, y ) ∈ P + the same alternating schemeoccurs, with opposite signs. In particular, the function ω ( t ) is a.s. negative for all t largeenough. In order to state our first result, which computes the a.s. limit velocity of ω ( t ) , letus finally introduce the parameters γ = ρα α ∈ (0 , /
2) and γ = (1 − ρ ) α α ∈ (0 , / . Theorem A.
Assume ρ ∈ (0 , and ( x, y ) (cid:54) = (0 , . Then, under P ( x,y ) , one has ω ( t )log t a.s. −→ − πγ ) sin( πγ ) α sin( π ( γ + γ )) as t → + ∞ . Note that in the Brownian motion case α = 2, we have ρ = 1 / γ = γ = 1 / , so that2 sin( πγ ) sin( πγ ) α sin( π ( γ + γ )) = √ · The constant −√ / π/ √ , as can be checked by an appropriate contour integration. The proof of Theorem A goesbasically along the same lines as in [8]. We consider the successive hitting times of 0 for theintegrated process X : T (1)0 = T = inf { t > , X t = 0 } and T ( n )0 = inf { t > T ( n − , X t = 0 } , which can be viewed as the half-winding times of Z. We first check that n (cid:55)→ T ( n )0 increasesa.s. to + ∞ as soon as ( x, y ) (cid:54) = (0 , . The exact exponential rate of escape of T ( n )0 , whichyields the exact winding velocity, is computed thanks to an elementary large deviation argu-ment involving the law of L T under P ( x, , a certain transform of the half-Cauchy distributionas observed in [10]. Notice that contrary to [8] where the proof is only sketched, we providehere an argument with complete details.In the Brownian case α = 2 , an expression of the law of the bivariate random variable( T ( n )0 , | L T ( n )0 | ) under P (0 ,y ) has been given in Theorem 1 of [7], in terms of the modified Besselfunction of the first kind. This expression becomes very complicated under P ( x,y ) when x (cid:54) = 0 , CHRISTOPHE PROFETA AND THOMAS SIMON even for n = 1 - see Formula (2) p.4 in [7]. In all cases, this expression is not informativeenough to evaluate the upper tails of T ( n )0 . In [9] it was shown that P ( x,y ) [ T ( n )0 ≥ t ] (cid:16) t − / (log t ) n − as t → + ∞ where, here and throughout, the notation f ( t ) (cid:16) g ( t ) means that there exist two constants0 < κ ≤ κ < + ∞ such that κ f ( t ) ≤ g ( t ) ≤ κ f ( t ) as t → + ∞ . On the other hand,Theorem A in our previous paper [10] shows the non-trivial asymptotics P ( x,y ) [ T > t ] (cid:16) t − θ as t → + ∞ for ( x, y ) ∈ P − , with θ = ρ/ (1 + α (1 − ρ )) . By symmetry, the latter result also shows that P ( x,y ) [ T > t ] (cid:16) t − θ as t → + ∞ for ( x, y ) ∈ P + , with θ = (1 − ρ ) / (1 + αρ ) . The second mainresult of this paper connects the two above estimates.
Theorem B.
Assume that ρ ∈ (0 , and ( x, y ) ∈ P − . For every n ≥ , the followingasymptotics hold as t → + ∞ : P ( x,y ) [ T ( n )0 > t ] (cid:16) t − θ (log t )[ n − ] if ρ < / , P ( x,y ) [ T ( n )0 > t ] (cid:16) t − θ (log t )[ n ] − if ρ > / , P ( x,y ) [ T ( n )0 > t ] (cid:16) t − θ (log t ) n − if ρ = 1 / . By symmetry, the same result holds for ( x, y ) ∈ P + with θ and θ switched. In the abovestatement, the separation of cases is intuitively clear, since for ρ < / ρ > / x, y ) ∈ P − . This is handled thanks to a uniform estimate on the Mellintransform of the harmonic measure P ( x,y ) [ L T ∈ . ] , and a general estimate on the upper tailsof the product of two positive independent random variables. These two estimates have bothindependent interest. 2. Proofs
Preliminary results.
As mentioned before, we first establish some estimates on theharmonic measure of the left half-plane with respect to the stable Kolmogorov process. Whenstarting from ( x, y ) ∈ P − , the process ( X, L ) ends up in exiting P − on the positive verticalaxis, and its exit distribution is given by the law of L T under P ( x,y ) . This distribution iscalled the harmonic measure since by the generalized Poisson formula, it allows to constructharmonic functions with respect to the degenerate operator L α,ρy + y ∂∂x on the half-plane, where L α,ρy is the generator of the stable L´evy process L. However, weshall not pursue these lines of research here.
INDINGS OF THE STABLE KOLMOGOROV PROCESS 5
Lemma 1.
Assume that ( x, y ) ∈ P − . The Mellin transform s (cid:55)→ E ( x,y ) [ L s − T ] is real-analyticon (1 / ( γ − , / (1 − γ )) , with two simple poles at / ( γ − and / (1 − γ ) . In particular, therandom variable L T has a smooth density f x,y under P ( x,y ) , and there exist c , c > suchthat f x,y ( z ) ∼ z → c z αθ/γ and f x,y ( z ) ∼ z → + ∞ c z − αθ − . Proof.
Observe first that the smoothness and the asymptotic behaviour of the density func-tion of L T are a direct consequence of the statement on the Mellin transform, thanks to theconverse mapping theorem stated e.g. as Theorem 4 in [5]. This latter statement is also adirect consequence of Theorem B in [10] when either x = 0 or y = 0. From now on we shalltherefore assume that xy (cid:54) = 0. By Proposition 2 (i) and Equation (3.2) in [10] we have E ( x,y ) (cid:2) L s − T (cid:3) = π (cid:90) + ∞ E ( x,y ) (cid:2) X − νt { X t > } (cid:3) dt (1 + α ) − ν (Γ (1 − ν )) Γ(1 − s ) sin( πs (1 − γ )) (2.1)with s = (1 − ν )(1 + α ) ∈ (0 , E ( x,y ) [ X − νt { X t > } ] = Γ(1 − ν ) π (cid:90) ∞ λ ν − e − c α,ρ λ α t α +1 sin( λ ( x + yt ) + s α,ρ λ α t α +1 + πν/ dλ for every ν ∈ (0 , , with s α,ρ = sin( πα ( ρ − / α + 1 ∈ ( − ,
1) and c α,ρ = cos( πα ( ρ − / α + 1 ∈ (0 , . For every β ∈ (0 , ν ) this yields (cid:90) −∞ | x | β − E ( x,y ) [ X − νt { X t > } ] dx = Γ(1 − ν ) π (cid:90) + ∞ λ ν − e − c α,ρ λ α t α +1 (cid:90) −∞ | x | β − sin( λ ( x + yt ) + s α,ρ λ α t α +1 + πν/ dx dλ = Γ(1 − ν )Γ( β ) π (cid:90) + ∞ λ ν − β − e − c α,ρ λ α t α +1 sin( λyt + s α,ρ λ α t α +1 + π ( ν − β ) / dλ = Γ(1 − ν )Γ( β )Γ(1 − ν + β ) E (0 ,y ) (cid:104) X − ν + βt { X t > } (cid:105) where the switching of the first equality is justified exactly as in Lemma 1 of [10], and thesecond equality follows from trigonometry and generalized Fresnel integrals - see (2.1) and CHRISTOPHE PROFETA AND THOMAS SIMON (2.2) in [10]. Assume first that y <
0. From Proposition 2 (ii) in [10], we obtain (cid:90) + ∞ (cid:90) −∞ | x | β − E ( x,y ) [ X − νt { X t > } ] dx dt = Γ(1 − ν )Γ( β )Γ(1 − ν + β ) π ( α +1) − ν + β Γ(1 − s − β (1+ α )) sin( πραβ + πγs ) | y | s + β (1+ α ) − for every β ∈ (0 , (1 − s ) / ( α + 1)) . Putting this together with (2.1), we finally deduce (cid:90) −∞ | x | β − E ( x,y ) (cid:2) L s − T (cid:3) dx = ( α + 1) β Γ( β ) sin( πραβ + πγs )Γ(1 − ν + β )Γ(1 − s − β (1 + α ))Γ(1 − ν )Γ(1 − s ) sin( πs (1 − γ )) | y | s + β (1+ α ) − . (2.2)We shall now invert this Mellin transform in the variable β in order to get a suitable integralexpression for E ( x,y ) [ L s − T ] . Fix β ∈ (0 , (1 − s ) / ( α + 1)) . On the one hand, since ρα <
1, wehave Γ( β ) cos(( πραβ + πγs ) /
2) = (cid:90) ∞ x β − e − x cos( πρα/ cos( x sin( πρα/
2) + πγs/ dx and Γ(1 − ν + β ) sin(( πραβ + πγs ) /
2) = Γ(1 − ν + β ) sin (cid:16) πρα β + 1 − ν ) (cid:17) = (cid:90) ∞ x β − x − ν e − x cos( πρα/ sin( x sin( πρα/ dx. On the other hand, a change of variable in the definition of the Gamma function shows thatΓ(1 − s − β (1 + α )) = 11 + α (cid:90) + ∞ x β − x s − α e − x − / (1+ α ) dx. Setting K s ( ξ ) = (cid:90) + ∞ z s α − e − cos( πρα/ ξ/z + z ) cos (cid:18) ξz sin( πρα/
2) + πγs/ (cid:19) sin( z sin( πρα/ dz, we can now invert (2.2) and obtain, applying Fubini’s theorem and using the notation (cid:101) x = x (1 + α ) | y | α , a new expression for the Mellin transform of L T : E ( (cid:101) x,y ) (cid:2) L s − T (cid:3) = 2 | y | s − | x | s − α (1 − s )Γ( s/ (1 + α ))Γ(2 − s ) sin( πs (1 − γ )) (cid:90) + ∞ ξ − s α − e − ( ξ | x | ) / (1+ α ) K s ( ξ ) dξ. Since − γ = 1 + αρα +(1 − αρ ) < ρ <
2, it remains to prove that the function H s ( x ) = (1 − s ) (cid:90) ∞ ξ − s α − e − ( ξ | x | ) / (1+ α ) K s ( ξ ) dξ INDINGS OF THE STABLE KOLMOGOROV PROCESS 7 admits an analytic continuation on [1 / ( γ − , / (1 − γ )]. Observe first that for any s > − − α ,the function K s is uniformly bounded on [0 , + ∞ ) by | K s ( ξ ) | ≤ sin( πρα/ (cid:90) + ∞ z s α e − cos( πρα/ z dz = sin( πρα/ πρα/ s α +1 Γ (cid:18) s α + 1 (cid:19) . As a consequence, the function H s has an analytic continuation on ( − − α, ⊃ [1 / ( γ − , γ − = − α α (1 − ρ ) > − − α . Next for every s ∈ (0 ,
1) an integration by parts showsthat H s ( x ) = (1 + α ) (cid:90) ∞ ξ − s α ddξ (cid:18) e − ( ξ | x | ) / (1+ α ) K s ( ξ ) (cid:19) dξ = (1 + α ) (cid:90) ∞ ξ − s α e − ( ξ | x | ) α K (cid:48) s ( ξ ) dξ − α | x | / (1+ α ) (cid:90) + ∞ ξ − s − α α e − ( ξ | x | ) α K s ( ξ ) dξ, where K (cid:48) s is well-defined on [0 , + ∞ ) for any s >
0, and bounded by | K (cid:48) s ( ξ ) | ≤ πρα/ (cid:90) ∞ z s α − e − cos( πρα/ z dz = 2 sin( πρα/ πρα/ s/ (1+ α ) Γ (cid:18) s α (cid:19) . Consequently, the function H s also admits an analytic continuation on (0 , ⊃ (0 , / (1 − γ )].This completes the proof in the case y < . The case y > (cid:3)
Our second preliminary result is elementary, but we could not find any reference in theliterature and we hence provide a proof.
Lemma 2.
Let µ ≥ ν > and n, p ∈ N . Assume that X and Y are two independent positiverandom variables such that : P [ X ≥ z ] (cid:16) z → + ∞ z − ν (log z ) n and P [ Y ≥ z ] (cid:16) z → + ∞ z − µ (log z ) p . Then P [ XY ≥ z ] (cid:16) z → + ∞ z − ν (log z ) n + p +1 if µ = ν, P [ XY ≥ z ] (cid:16) z → + ∞ z − ν (log z ) n if µ > ν. Proof.
We first decompose the product as P [ XY ≥ z ] = (cid:90) ∞ P [ X ≥ zy − ] P [ Y ∈ dy ] . Therefore, for z > A large enough, P [ XY ≥ z ] ≥ (cid:90) √ zA P [ X ≥ zy − ] P [ Y ∈ dy ] ≥ κ z ν (cid:90) √ zA y ν (log( zy − )) n P [ Y ∈ dy ] ≥ κ (log( z )) n n z ν (cid:90) √ zA y ν P [ Y ∈ dy ] . CHRISTOPHE PROFETA AND THOMAS SIMON
Then, integrating by parts, (cid:90) √ zA y ν P [ Y ∈ dy ] = A ν P [ Y ≥ A ] − z ν/ P [ Y ≥ √ z ] + ν (cid:90) √ zA y ν − P [ Y ≥ y ] dy. Now, if ν < µ , this expression remains bounded as z → + ∞ . Assume therefore that µ = ν .In this case, we have : (cid:90) √ zA y ν P [ Y ∈ dy ] ≥ A ν P [ Y ≥ A ] − κ p (log z ) p + νκ (cid:90) √ zA (log y ) p y dy ∼ z → + ∞ νκ (log z ) p +1 p +1 ( p + 1) , which gives the lower bound. To obtain the upper bound, we separate the integral in threeparts and proceed similarly, with ε small enough: (cid:90) /ε P [ X ≥ zy − ] P [ Y ∈ dy ] + (cid:90) εz /ε P [ X ≥ zy − ] P [ Y ∈ dy ] + (cid:90) ∞ εz P [ X ≥ zy − ] P [ Y ∈ dy ] ≤ P [ X ≥ εz ] + κ (log( εz )) n z ν (cid:90) εz /ε y ν P [ Y ∈ dy ] + P [ Y ≥ εz ]and the proof is concluded as before, using an integration by parts and looking separatelyat both cases ν < µ and ν = µ . (cid:3) Proof of Theorem A.
By symmetry, it is enough to show Theorem A for ( x, y ) ∈ P − . Consider the sequence (cid:16) T ( n )0 , | L T ( n )0 | (cid:17) n ≥ and set {F n , n ≥ } for its natural completed filtration. It is easy to see from the strongMarkov and scaling properties of Z that this sequence is Markovian. To be more precise,starting from P − and taking into account the possible asymmetry of the process L , we havethe following identities for all p ≥ . (cid:16) T (2 p )0 , | L T (2 p )0 | (cid:17) d = (cid:16) T (2 p − + | L T (2 p − | α τ + , | L T (2 p − | (cid:96) + (cid:17) with ( τ + , (cid:96) + ) ⊥ F p − distributed as ( T , | L T | ) under P (0 , , and (cid:16) T (2 p +1)0 , | L T (2 p +1)0 | (cid:17) d = (cid:16) T (2 p )0 + | L T (2 p )0 | α τ − , | L T (2 p )0 | (cid:96) − (cid:17) with ( τ − , (cid:96) − ) ⊥ F p distributed as ( T , | L T | ) under P (0 , − . The starting term ( T , | L T | ) hasthe same law as ( τ − , (cid:96) − ) if x = 0 and y = − . By induction we deduce the identities | L T (2 p )0 | d = | L T | × p − (cid:89) k =1 (cid:96) − k × p (cid:89) k =1 (cid:96) + k and | L T (2 p +1)0 | d = | L T | × p (cid:89) k =1 (cid:96) − k × p (cid:89) k =1 (cid:96) + k where, here and throughout, ( τ ± k , (cid:96) ± k ) k ≥ are two i.i.d. sequences distributed as ( τ ± , (cid:96) ± ) , and all products are assumed independent. From Theorem B (i) in [10] and its symmetric INDINGS OF THE STABLE KOLMOGOROV PROCESS 9 version, the Mellin tranforms of (cid:96) ± are given by E (cid:2) ( (cid:96) − ) s − (cid:3) = sin( πγs )sin( π (1 − γ ) s ) and E (cid:2) ( (cid:96) + ) s − (cid:3) = sin( πγs )sin( π (1 − γ ) s ) (2.3)for each real s in the respective domain of definition, which is in both cases an open intervalcontaining 1. This entails that E [ | log( (cid:96) ± ) | ] < + ∞ , with E (cid:2) log( (cid:96) − ) (cid:3) = π cot( πγ ) > E (cid:2) log( (cid:96) + ) (cid:3) = π cot( πγ ) > . (2.4)The following lemma is intuitively obvious. Lemma 3.
Assume ( x, y ) ∈ P − . Then one has T ( n )0 → + ∞ and the process Z never hitsthe origin, a.s. under P ( x,y ) . Proof.
To prove the first statement, it is enough to show that S n = T ( n )0 − T ( n − → + ∞ a.s.as n → ∞ . Set κ α,ρ = πα πγ ) + cot( πγ )) = πα sin( π ( γ + γ ))2 sin( πγ ) sin( πγ ) > . From the above discussion, we have S p d = | L T | α × τ + × (cid:32) p − (cid:89) k =1 (cid:96) − k × (cid:96) + k (cid:33) α (2.5)for every p ≥ , with independent products on the right-hand side. For every ε ∈ (0 , κ α,ρ ) , this entails P ( x,y ) (cid:2) S p ≤ e p − κ α,ρ − ε ) (cid:3) ≤ P ( x,y ) (cid:2) | L T | ≤ e − ε ( p − / α (cid:3) + P (cid:2) τ + ≤ e − ε ( p − / (cid:3) + P (cid:34) p − p − (cid:88) k =1 log( (cid:96) − k ) ≤ π cot( πγ ) − ε/ (cid:35) + P (cid:34) p − p − (cid:88) k =1 log( (cid:96) + k ) ≤ π cot( πγ ) − ε/ (cid:35) . From Lemma 1, there exists θ ( ε ) > P ( x,y ) (cid:2) | L T | ≤ e − ε ( p − / α (cid:3) < e − pθ ( ε ) for p large enough. On the other hand, we have P (cid:2) τ + ≤ e − ε ( p − / (cid:3) ≤ P (0 , (cid:2) inf { L t , t ≤ e − ε ( p − / } < (cid:3) = P (0 , (cid:104) sup { ˆ L t , t ≤ } > e ε ( p − / α (cid:105) ≤ e − pθ ( ε ) for some θ ( ε ) > p large enough, where we have set ˆ L = − L, the equality followingfrom translation invariance and self-similarity, and the second inequality from the generalestimate of Theorem 12.6.1 in [11]. Last, the existence of some θ ( ε ) > e − pθ ( ε ) for p large enough is a standardconsequence of (2.3), (2.4) and Cram´er’s theorem - see e.g. Theorem 1.4 in [3], recalling that the assumption (I.5) can be replaced by (I.17) therein. We can finally appeal to theBorel-Cantelli lemma to deduce, having let ε → , lim inf p →∞ p log( S p ) ≥ κ α,ρ > S p → + ∞ a.s. and an entirely similar argument yields S p +1 → + ∞ a.s.This concludes the proof of the first part of the lemma.The second part is easier. If α ≤ , it is well-known that L never hits zero, so that Z never hits the origin. If α > , we see from Lemma 1 that L T has no atom at zero under P ( x,y ) and because (cid:96) ± are absolutely continuous, all L T ( n )0 ’s have no atom at zero. We finallyget P ( x,y ) [ Z visits the origin] = P ( x,y ) (cid:34) (cid:91) n ≥ (cid:110) L T ( n )0 = 0 (cid:111)(cid:35) = 0where the first identification comes from the fact that T ( n )0 → + ∞ a.s. (cid:3) We can now finish the proof of Theorem A. Set θ = (cid:92) Z Z T ∈ ( − π,
0) a.s. Observing as in[8] the a.s. identifications { ω ( t ) ≥ − ( n − π + θ } = { T ( n )0 ≥ t } and { ω ( t ) ≤ − ( n − π + θ } = { T ( n − ≤ t } , we see that Theorem A amounts to show that1 n log( T ( n )0 ) a.s. −→ πα sin( π ( γ + γ ))2 sin( πγ ) sin( πγ ) = κ α,ρ as n → + ∞ . Firstly, with the above notation, we have a.s. under P ( x,y ) lim inf n →∞ n log( T ( n )0 ) ≥ lim inf n →∞ n log( S n ) ≥ κ α,ρ , where the second inequality comes from (2.6) and its analogue for n odd. To obtain theupper bound, we will proceed as in the above Lemma 3. Fixing ε > , we have P ( x,y ) (cid:104) T ( n )0 ≥ e n ( κ α,ρ + ε ) (cid:105) ≤ n (cid:88) k =1 P ( x,y ) (cid:2) S k ≥ n − e n ( κ α,ρ + ε ) (cid:3) ≤ n (cid:88) k =1 P ( x,y ) (cid:2) S k ≥ e n ( κ α,ρ + ε/ (cid:3) for n large enough, with the above notation for S k and having set S = T . Recalling (2.5)we have for every k = 2 p ≤ n P ( x,y ) (cid:2) S p ≥ e n ( κ α,ρ + ε/ (cid:3) ≤ P ( x,y ) (cid:2) | L T | ≥ e nε/ α (cid:3) + P (cid:2) τ + ≥ e nε/ (cid:3) + P (cid:34) n p − (cid:88) k =1 log( (cid:96) − k ) ≥ π cot( πγ ) + ε/ (cid:35) + P (cid:34) n p − (cid:88) k =1 log( (cid:96) + k ) ≥ π cot( πγ ) + ε/ (cid:35) . INDINGS OF THE STABLE KOLMOGOROV PROCESS 11
Again from Lemma 1, there exists θ ( ε ) > P ( x,y ) (cid:2) | L T | ≥ e nε/ α (cid:3) < e − nθ ( ε ) for n large enough, whereas P (cid:2) τ + ≥ e nε/ (cid:3) ≤ P (0 , (cid:104) sup { ˆ L t , t ≤ } < e − nε/ α (cid:105) ≤ e − nθ ( ε ) for some θ ( ε ) > n large enough, the second inequality following e.g. from PropositionVIII.2 in [1]. To handle the third term, we separate according as p ≤ √ n or p > √ n. In thefirst case, we have the upper bound P (cid:34) n p − (cid:88) k =1 log( (cid:96) − k ) ≥ π cot( πγ ) + ε/ (cid:35) ≤ √ n (cid:88) k =1 P (cid:20) √ n log( (cid:96) − k ) ≥ π cot( πγ ) + ε/ (cid:21) ≤ e − θ ( ε ) √ n for some θ ( ε ) > n large enough, using Lemma 1 for the second inequality. In thesecond case, applying Cram´er’s theorem exactly as in Lemma 3 gives the upper bound P (cid:34) n p − (cid:88) k =1 log( (cid:96) − k ) ≥ π cot( πγ ) + ε/ (cid:35) ≤ e − θ ( ε ) √ n for some θ ( ε ) > n large enough. The fourth term is estimated in the same way andwe finally get the existence of some θ ( ε ) > P ( x,y ) (cid:2) S p ≥ e n ( κ α,ρ + ε/ (cid:3) ≤ e − θ ( ε ) √ n for n large enough. An analogous estimate is obtained for P ( x,y ) (cid:2) S p +1 ≥ e n ( κ α,ρ + ε/ (cid:3) and wecan apply as usual the Borel-Cantelli lemma to show the required upper boundlim sup n →∞ n log( T ( n )0 ) ≤ κ α,ρ + ε, for all ε > , a.s. under P ( x,y ) . (cid:3) Proof of Theorem B.
Recall the decomposition T ( n )0 = S + · · · + S n with S d = T ,S p d = | L T | α × τ + × (cid:32) p − (cid:89) k =1 (cid:96) − k × (cid:96) + k (cid:33) α , S p +1 d = | L T | α × τ − × (cid:32) p − (cid:89) k =1 (cid:96) − k × p (cid:89) k =1 (cid:96) + k (cid:33) α , and the above notation for ( τ ± , (cid:96) ± ) . Let us first investigate the upper tails of the distributionof each S k under P ( x,y ) . We know that P ( x,y ) [ T > t ] (cid:16) P ( x,y ) [ | L T | α > t ] (cid:16) P [ τ − > t ] (cid:16) P [( (cid:96) − ) α > t ] (cid:16) t − θ and P [ τ + > t ] (cid:16) P [( (cid:96) + ) α > t ] (cid:16) t − θ as t → + ∞ . Supposing first ρ = 1 / θ = θ, a successive application of Lemma 2 showsthat P ( x,y ) [ S k > t ] (cid:16) t − θ (log t ) k − as t → + ∞ for every k ≥ . Suppose then ρ < / θ < θ, we obtain in a similar way P ( x,y ) [ S p > t ] (cid:16) t − θ (log t ) p − and P ( x,y ) [ S p +1 > t ] (cid:16) t − θ (log t ) p as t → + ∞ , for every p ≥ . Last, if ρ > / P ( x,y ) [ S p > t ] (cid:16) t − θ (log t ) p − and P ( x,y ) [ S p +1 > t ] (cid:16) t − θ (log t ) p − as t → + ∞ , for every p ≥ . All in all, for all k ≥ , this shows that P ( x,y ) [ S k > t ] (cid:16) t − θ (log t )[ k − ] if ρ < / P ( x,y ) [ S k > t ] (cid:16) t − θ (log t )[ k ] − if ρ > / P ( x,y ) [ S k > t ] (cid:16) t − θ (log t ) k − if ρ = 1 / , and we also know that P ( x,y ) [ S > t ] (cid:16) t − θ . The immediate estimate P ( x,y ) [ T ( n )0 > t ] ≥ P ( x,y ) [ S n > t ] yields the required lower bound. To get the upper bounds, it suffices to write P ( x,y ) [ T ( n )0 > t ] ≤ n (cid:88) k =1 P ( x,y ) [ S k > t/n ]and to control the sum separately according as ρ < / , ρ > / ρ = 1 / . We leave thedetails to the reader. (cid:3)
Acknowledgement.
Ce travail a b´en´efici´e d’une aide de la Chaire
March´es en Mutation ,F´ed´eration Bancaire Fran¸caise.
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INDINGS OF THE STABLE KOLMOGOROV PROCESS 13
Laboratoire de Math´ematiques et Mod´elisation d’Evry (LaMME), Universit´e d’Evry-Val-d’Essonne, UMR CNRS 8071, F-91037 Evry Cedex.
Email : [email protected] Laboratoire Paul Painlev´e, Universit´e Lille 1, F-59655 Villeneuve d’Ascq Cedex andLaboratoire de Physique Th´eorique et Mod`eles Statistiques, Universit´e Paris-Sud, F-91405Orsay Cedex.
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