On Wiener-Hopf factors for stable processes
aa r X i v : . [ m a t h . P R ] J a n On Wiener-Hopf factors for stable processes
Piotr Graczyk ∗ LAREMA, Universit´e d’Angers, 2 Bd Lavoisier,49045 Angers Cedex 1, [email protected]
Tomasz Jakubowski † LAREMA, Universit´e d’Angers, 2 Bd Lavoisier,49045 Angers Cedex 1, FranceInstitute of Mathematics, Wroc law University of Technology, Wyb. Wyspia´nskiego 27,50-370 Wroc law, [email protected]
Abstract
We give a series representation of the logarithm of the bivariateLaplace exponent κ of α -stable processes for almost all α ∈ (0 , R´esum´e
Nous donnons un d´eveloppement en s´erie du logarithme de l’expo-sant de Laplace bivari´e κ des processus α -stables pour presque tous α ∈ (0 , MSC:
Keywords: stable process, Wiener-Hopf factorization ∗ This research was partially supported by grants MNiSW N N201 373136 and ANR-09-BLAN-0084-01. † This research was partially supported by grants MNiSW 3971/B/H03/2009/37, ANR-09-BLAN-0084-01 and the fellowship of CCRRDT Pays de la Loire. Introduction
The fluctuation theory of L´evy processes is one of the domains of probabil-ity very actively developing in the last years, and with important applicationsin mathematical finance; cf. the recent monograph of A. Kyprianou [12] andpapers [2], [5], [6], [13]. The α -stable L´evy processes play a primordial rolein this theory. We address in this article one of the key problems of theWiener-Hopf factorization theory of the α -stable processes: the computationof the bivariate Laplace exponent κ ( γ, β ).The aim of this paper is to give a series representation of the integral g ( β ) = sin( πρ ) π Z ∞ β log(1 + x α ) x + 2 xβ cos( πρ ) + β dx (1)for almost all α ∈ (0 ,
2] and ρ ∈ [1 − /α, /α ] ∩ (0 , κ ( γ, β ) of the ascending ladder process builtfrom the α -stable process X t with index of stability α and ρ = P ( X > γ ρ exp (cid:8) g ( βγ − /α ) (cid:9) = κ ( γ, β )= k exp (cid:26)Z ∞ Z (0 , ∞ ) e − t − e − γt e − βx t P ( X t ∈ dx ) dt (cid:27) . The integral (1) was introduced by Darling in [7] for ρ = 1 / α = 1 and ρ = 1 /
2, which corresponds to the symmetricCauchy process and later by Bingham [4] for spectrally negative stable pro-cesses (1 /ρ = α ∈ (1 , α, ρ ) satisfying ρ + k = l/α for some k ∈ N ∪ { } and l ∈ N . Although thefunction κ plays an important role in the theory of stable (in general L´evy)processes, the only known closed expression for it is due to Doney. In thisnote we expand the function g to a power series for almost all α and ρ . Wedenote by L the set of Liouville numbers, which will be defined in Section 2.Let A = (0 , \ ( Q ∪ L ). We note that if α ∈ A then by Lemma 2, 1 /α ∈ A .The main result of this paper is Theorem 1.
Let α ∈ A , ρ ∈ [1 − /α, /α ] ∩ (0 , and < β < . Then g ( β ) = ∞ X m =1 ( − m +1 β m sin( ρmπ ) m sin( mπα ) + ∞ X k =1 ( − k +1 β αk sin( ραkπ ) k sin( αkπ ) . (2)2e note that in view of Lemma 5 it suffices to consider only 0 < β < α ∈ L ∩ (0 , L has a Lebesgue measure 0 hence A contains almost all α ∈ (0 , α (Proposition 10) but the expressionis not so closed as in Theorem 1. If ρ + k = l/α for some integers k and l theformula (2) may be simplified, in particular one may obtain results achievedby Doney in [8] (see Remark 1).The formula (2) opens a way to applications for the study of various func-tionals of an α -stable L´evy process, in particular of the long time behaviorof the supremum process or the law of the first passage time, cf. the recentresults of Bernyk, Dalang and Peskir [2] and Kuznetsov [11]. We also profitfrom the Theorem 1 in a forthcoming work [9], devoted to the first passagetime of symmetric stable processes.The paper is organized as follows. In Section 2 we define Liouville num-bers and prove some auxiliary lemmas. In Section 3 we prove the mainTheorem 1. In Section 4 we give some remarks, applications and examples. A number x ∈ R is called a Liouville number if it may be well approx-imated by rational numbers. More precisely for any n ∈ N there exist in-finitely many pairs of integers p, q such that (see e.g. [1])0 < (cid:12)(cid:12)(cid:12)(cid:12) x − pq (cid:12)(cid:12)(cid:12)(cid:12) < q n . We denote by L the set of all Liouville numbers. First we note Lemma 2. x ∈ L if and only if /x ∈ L . The proof does not seem available in the literature. The following proofwas proposed by M. Waldschmidt ([14]).
Proof.
Let x
6∈ L . There are c ∈ R and d ∈ N such that for all p ∈ Z , q ∈ N , (cid:12)(cid:12)(cid:12)(cid:12) x − pq (cid:12)(cid:12)(cid:12)(cid:12) ≥ cq d . p ∈ Z , q ∈ N . We may and do suppose that | /x − p/q | <
1. Hence | p | /q < ( | x | + 1) / | x | and (cid:12)(cid:12)(cid:12)(cid:12) x − pq (cid:12)(cid:12)(cid:12)(cid:12) = | p | q | x | (cid:12)(cid:12)(cid:12)(cid:12) x − qp (cid:12)(cid:12)(cid:12)(cid:12) ≥ | p | q | x | c | p | d ≥ | x | d − (1 + | x | ) d − cq d . Lemma 3.
For any x ∈ A and β ∈ (0 , we have ∞ X m =1 β m | sin( mxπ ) | < ∞ . Proof.
Since x ∈ R \ ( Q ∪ L ), there is N ∈ N such that (cid:12)(cid:12)(cid:12) x − pq (cid:12)(cid:12)(cid:12) > q N for allintegers p, q >
0. Hence | sin( mxπ ) | > m N − and the lemma follows.In the sequel we will need following formulas taken from [10] (formulas1.445.7, 1.422.3, 1.353.1) ∞ X m =1 ( − m +1 m sin( mz ) m − w = π zw )sin( wπ ) , z ∈ ( − π, π ) , w ∈ R \ Z , (3) π sin( πz ) = 1 z − ∞ X k =1 ( − k zk − z , z ∈ R \ Z , (4) n − X k =1 p k sin( kx ) = p sin( x ) − p n sin( nx ) + p n +1 sin(( n − x )1 − p cos( x ) + p . (5) Lemma 4.
Let α ∈ A and ρ ∈ [1 − /α, /α ] ∩ (0 , . Then there areconstants C and N such that for all M, k ∈ N M X m =1 ( − m sin( mρπ ) mm − ( αk ) ≤ Ck N . Proof.
Let K be the smallest integer larger then αk + 1. Like in the proofof Lemma 3 we take N such that (cid:12)(cid:12)(cid:12) α − pq (cid:12)(cid:12)(cid:12) > q N for all integers p, q . Then | m − ( αk ) | > mk − N +1 for all m, k ∈ N and we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K − X m =1 ( − m sin( mρπ ) mm − ( αk ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( αk + 1) k N − ≤ k N . a m = ( − m sin( mρπ ) and b m = mm − ( αk ) . By (5) for any M ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X m =1 a m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ρπ )) = c . Since b m is decreasing for m ≥ K we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X m = K a m b m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M − X m = K ( b m − b m +1 ) m X n = K a n + b M M X n = K a n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M − X m = K ( b m − b m +1 ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X n = K a n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + b M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X n = K a n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ cb K ≤ c. The following lemma justifies our restriction in Theorem 1 to 0 < β < Lemma 5. κ (1 , β ) = κ (1 , /β ) β αρ . Proof.
After substituting x = 1 /y we get g ( β ) = β sin( πρ ) π Z ∞ log(1 + y α ) − log( y α )1 + 2 yβ cos( πρ ) + y β dy = g (1 /β ) + α sin( πρ ) π Z ∞ log( β ) − log( z )1 + 2 z cos( πρ ) + z dz = g (1 /β ) + α log( β ) π Z ∞ sin( πρ )1 + 2 z cos( πρ ) + z dz = g (1 /β ) + αρ log( β )and the lemma follows.A derivative of the function g is equal to g ′ ( β ) = ∂∂β sin( πρ ) π Z ∞ log(1 + β α x α ) x + 2 x cos( πρ ) + 1 dx = sin( πρ ) απ Z ∞ x α x α x + 2 xβ cos( πρ ) + β dx . Our aim is to prove 5 emma 6.
Let α ∈ A , ρ ∈ [1 − /α, /α ] ∩ (0 , and < β < . Then g ′ ( β ) = ∞ X m =1 ( − m +1 β m − sin( ρmπ )sin( mπα ) + α ∞ X k =1 ( − k +1 β αk − sin( ραkπ )sin( αkπ ) . Lemma 7.
For any p > and < b < Z b y p y dy = ∞ X k =0 ( − k b k +1+ p k + 1 + p . Proof.
By Fubini theorem Z b y p y dy = Z b ∞ X k =0 ( − k y p + k dy = ∞ X k =0 ( − k b k +1+ p k + 1 + p . Lemma 8.
For any < b ≤ and p ∈ (0 , ∞ ) \ N we have Z ∞ b y − p y dy = π sin( pπ ) + ∞ X k =0 ( − k +1 b k +1 − p k + 1 − p . (6) Proof.
Since the derivatives in b of both sides of (6) are equal we have for b ∈ (0 , Z ∞ b y − p y dy = C + ∞ X k =0 ( − k +1 b k +1 − p k + 1 − p . To calculate the constant C we take b → C = Z ∞ y − p y dy − ∞ X k =0 ( − k +1 k + 1 − p = Z x p − x dx − ∞ X k =0 ( − k +1 k + 1 − p = ∞ X k =0 ( − k Z x p + k − dx − ∞ X k =0 ( − k +1 k + 1 − p = ∞ X k =0 ( − k k + p + ∞ X k =0 ( − k k + 1 − p = 1 p − ∞ X k =1 ( − k pk − p = π sin( pπ ) . n ∈ N lim p → n (cid:18) π sin( pπ ) + ( − n b n − p n − p (cid:19) = ( − n ln b , we get Corollary 9.
For p ∈ N and < b < Z ∞ b y − p y dy = ( − p ln b + X k ∈ N ,k = p − ( − k +1 b k +1 − p k + 1 − p . Proof of Lemma 6.
We note that (see [10, 1.447.1]) ∞ X m =0 ( − m x m sin(( m + 1) z ) = sin( z ) x + 2 x cos( z ) + 1 , | x | < . (7)First we will calculate R β . From (5) we deduce n − X k =1 ( − k p k sin( kz ) = − p sin( z ) − ( − n p n ( p sin(( n − z ) + sin( nz ))1 + 2 p cos( z ) + p Thus for any M ≥ z ∈ (0 , π ) and x ∈ (0 , β ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X m =0 ( − m sin(( m + 1) z ) (cid:18) xβ (cid:19) m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β (cid:0) xβ (cid:1) M +1 ( − M (cid:16) xβ sin( z ( M + 1)) + sin( z ( M + 2)) (cid:17) + sin( z ) x + 2 xβ cos( z ) + β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < z ) . α sin ρπ Z β x α x α β + 2 xβ cos( ρπ ) + x dx = α sin ρπ Z β x α x α β (( xβ ) + 2 xβ cos( ρπ ) + 1) dx = α Z β x α x α ∞ X m =0 β − ( − m sin( ρ ( m + 1) π ) (cid:18) xβ (cid:19) m dx = α Z β lim M →∞ M X m =0 x α x α β − ( − m sin( ρ ( m + 1) π ) (cid:18) xβ (cid:19) m dx = ∞ X m =0 ( − m β − − m sin( ρ ( m + 1) π ) Z β αx α + m x α dx . By Lemma 7 Z β αx α + m x α dx = Z β α y ( m +1) /α y dy = ∞ X k =0 ( − k β α ( k +1)+ m +1 k + 1 + ( m + 1) /α . Consequently α sin ρπ Z β x α x α β + 2 xβ cos( ρπ ) + x dx = α ∞ X m =0 ∞ X k =0 ( − k + m β α ( k +1) − sin( ρ ( m + 1) π ) α ( k + 1) + ( m + 1) . (8)Now we calculate R ∞ β . Similarly by the dominated convergence theorem we8et α sin( ρπ ) Z ∞ β x α x α β + 2 xβ cos( ρπ ) + x dx = α sin( ρπ ) Z ∞ β x α x α x (1 + 2 βx cos( ρπ ) + ( βx ) ) dx = α Z ∞ β x α − x α ∞ X m =0 ( − m (cid:18) βx (cid:19) m sin( ρ ( m + 1) π ) dx = ∞ X m =0 ( − m β m sin( ρ ( m + 1) π ) Z ∞ β αx α − − m x α dx = ∞ X m =0 ( − m β m sin( ρ ( m + 1) π ) Z ∞ β α y − (1+ m ) /α y dy . Since α ∈ A by Lemma 8 we get Z ∞ β α y − (1+ m ) /α y dy = π sin( m +1 α π ) − ∞ X k =0 ( − k β α ( k +1) − ( m +1) k + 1 − ( m + 1) /α . Therefore by Lemma 3 α sin ρπ Z ∞ β x α x α β + 2 xβ cos( ρπ ) + x dx = π ∞ X m =0 ( − m β m sin( ρ ( m + 1) π )sin( m +1 α π ) (9) − α ∞ X m =0 ∞ X k =0 ( − k + m β α ( k +1) − sin( ρ ( m + 1) π ) α ( k + 1) − ( m + 1) . Hence by (8), (9) and (3) we get1 π Z ∞ x α x α α sin ρπβ + 2 xβ cos( ρπ ) + x dx = ∞ X m =0 ( − m β m sin( ρ ( m + 1) π )sin( m +1 α π )+ 2 απ ∞ X m =0 ∞ X k =0 ( − k + m β α ( k +1) − ( m + 1) sin( ρ ( m + 1) π )( m + 1) − ( α ( k + 1)) = ∞ X m =0 ( − m β m sin( ρ ( m + 1) π )sin( m +1 α π ) + α ∞ X k =0 ( − k β α ( k +1) − sin( αρ ( k + 1) π )sin( α ( k + 1) π ) . Remark 1.
Put g k ( a, x ) = ∞ X m =1 x m U k − (cos( mπa )) m , (10)where U k ( x ) are the Chebyshev polynomials of the second type (we put U − ≡ ρ + k = l/α (like in [8]), l ≥ k ≥ α ∈ (0 , g ( β ) = g k ( α, ( − l +1 β α ) − g l (1 /α, ( − k +1 β ) , (11)We note that sums above correspond to the function f k defined in [8]. Proof.
First suppose α ∈ A and ρ = l/α − k . Since U k (cos( x )) = sin(( k +1) x )sin x we get (11) for all α ∈ A . Now for α ∈ (0 , \ A we take A ∋ α n → α and ρ n = l/α n − k . Passing to the limit we get (11) for α ∈ (0 , U k − (cos( z )) = m P n =0 cos((2 n + 1) z ) for k = 2 m + 2 , m P n =1 cos(2 nz ) for k = 2 m + 1 , ∞ X m =1 x m cos( mz ) m = − log( x − x cos( z ) + 1) , the functions g k ( a, x ) may be expressed by finite sums − g k ( a, x ) = k/ − P n =0 log( x − x cos((2 n + 1) aπ ) + 1) for even k , log(1 − x ) + ( k − / P n =1 log( x − x cos(2 naπ ) + 1) for odd k . xample 1. Let k = l = 1 then α ∈ (0 ,
1) and g ( β ) = − ∞ X m =1 β m /m + ∞ X m =1 β αm /m = − log(1 − β α ) + log(1 − β ) . Hence κ (1 , β ) = ˜ C − β − β α .A first application of Theorem 1 is to obtain new expressions for thefunctions g ′ ( β ) , g ( β ) and consequently κ (1 , β ) and κ ( γ, β ) for the values of β not concerned by the results of [8]. Proposition 10.
Let α ∈ Q ∩ (0 , and β ∈ (0 , . Then g ′ ( β ) = ∞ X m =1 ,mα N ( − m +1 β m − sin( ρmπ )sin( mπα ) + α ∞ X k =1 ,αk N ( − k +1 β αk − sin( ραkπ )sin( αkπ )+ α log( β ) π ∞ X m =1 ,mα ∈ N ( − m + mα β m − sin( ρmπ ) (12)+ αρ ∞ X k =1 ,αk ∈ N ( − k ( α +1) β αk − cos( αρkπ ) . Proof.
Let α = pq . Like in Remark 1 we take A ∋ α j = pq + √ j . We obtainresult by passing to the limit j → ∞ in the expression ∞ X m =1 ,mα N ( − m +1 β m − sin( ρmπ )sin( mπα j ) + α j ∞ X k =1 ,αk N ( − k +1 β α j k − sin( ρα j kπ )sin( α j kπ )+ ∞ X m =1 ,mα ∈ N ( − m +1 β m − sin( ρmπ )sin( mπα j ) + α j ∞ X k =1 ,αk ∈ N ( − k +1 β α j k − sin( ρα j kπ )sin( α j kπ )By Lemma 3 we pass with limit under sum signs. The first two terms obvi-ously converge to the first two terms in (12). If we take m = np , k = nq , the11econd line is equal to ∞ X n =1 (cid:18) ( − np +1 β np − sin( ρnpπ )sin( npπ/α j ) + α j ( − nq +1 β nqα j − sin( ρnqα j π )sin( nqα j π ) (cid:19) j →∞ −→ ∞ X n =1 ( − nq +1 ( − β ) np − p ( πρ cos( npπρ ) + log( β ) sin( npπρ )) πq (13)and the assertion of the proposition holds. For the detailed proof of (13) werefer to Appendix. Remark 2.
In fact Proposition 10 holds for α ∈ A ∪ ( Q ∩ (0 , Example 2.
Let α = 1 /
2. Then p = 1 and q = 2. We get g ′ ( β ) = 12 ∞ X k =0 ( − k β k − sin( ρ ( k + ) π )+ 12 π ∞ X n =1 ( − n β n − ( ρπ cos( nρπ ) + log( β ) sin( nρπ ))= (1+ β ) cos(( πρ ) / √ β − ρ ( β +cos( πρ ))2 − log( β ) sin( πρ ) π β + 2 β cos( πρ ) + 1Analogous simple expressions can be given for other rational α not coveredby the results of [8].Further applications of formula (2) from Theorem 1 are planned in theforthcoming paper [9] where symmetric α -stable processes X t in R are con-sidered. The starting point is the formula (see [12]) Z ∞ Z ∞ e − ηt e − θx E x ( e − γX t ; τ > t ) dt dx = 1( θ + γ ) κ ( η, γ ) κ ( η, θ ) , (14)where τ = τ (0 , ∞ ) is the first exit time from (0 , ∞ ) of the process X t .A better knowledge of κ then permits to get from (14) more informationabout the law of τ . 12 Appendix
Here we give a detailed proof of (13).
Lemma 11.
Let α j = pq + √ j . We have ∞ X n =1 (cid:18) ( − np +1 β np − sin( ρnpπ )sin( npπ/α j ) + α j ( − nq +1 β nqα j − sin( ρnqα j π )sin( nqα j π ) (cid:19) j →∞ −→ ∞ X n =1 ( − nq +1 ( − β ) np − p ( πρ cos( npπρ ) + log( β ) sin( npπρ )) πq (15) Proof.
Let us call F ( n, j ) = ( − np +1 sin( ρnpπ )sin( npπ/α j ) + α j ( − nq +1 sin( ρnpπ )sin( nqα j π ) F ( n, j ) = α j ( − nq +1 (sin( ρnqα j π ) − sin( ρnpπ ))sin( nqα j π ) F ( n, j ) = α j ( − nq +1 sin( ρnqα j π )sin( nqα j π ) ( β nqα j − np − Term by term convergence:
We note thatsin( npπα j ) = ( − nq +1 sin( nq √ πpj + √ ) , sin( nqα j π ) = ( − np sin( nq √ πj ) . Hence for fixed n and large j we have (cid:12)(cid:12)(cid:12)(cid:12) F ( n, j )sin( ρnpπ ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin( nq √ πj ) − pj + √ qqj sin( nq √ πpj + √ q )sin( nq √ πj ) sin( nq √ πpj + √ q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ P k =1 1(2 k +1)! (cid:18)(cid:16) nq √ πj (cid:17) k +1 + pq (cid:16) nq √ πpj + √ q (cid:17) k +1 (cid:19) nq √ π j nq √ π pj + √ q ) ≤ Knj j →∞ −→ , K is some constant independent of n and j . Therefore lim j →∞ F ( n, j ) =0. Furtherlim j →∞ F ( n, j ) = lim j →∞ α j ( − n ( p + q )+1 sin( ρnq √ π j ) cos( ρnπ ( qα j + p ) / nq √ π/j )= ( − n ( p + q )+1 pρ cos( ρnpπ ) q . Similarlylim j →∞ F ( n, j ) = lim j →∞ α j ( − n ( p + q )+1 sin( ρnqα j π ) ( β nq √ /j − nq √ π/j )= ( − n ( p + q )+1 p log( β ) sin( npπρ ) πq Uniform integrability with respect to the measure µ = P ∞ n =1 β np − δ n :We will show that for each k = 1 , , j ∈ N ∞ X n =1 | F k ( n, j ) | β np − < ∞ , and for every ε > δ > j ∈ N X n ∈ G | F k ( n, j ) | β np − < ε , whenever µ ( G ) < δ .From part 1) of the proof we see that for n < j/ (2 q √ F k ( n, j ) 3. Denote G j = { m ∈ N : m < j/ (2 q √ k ∈ N be the closest integer to nq √ /j . Then bydiophantine approximation | sin( nqα j π ) | = | sin(( k − nq √ /j ) π ) | ≥ | k − nq √ /j | nq j (cid:12)(cid:12)(cid:12)(cid:12) √ − kjnq (cid:12)(cid:12)(cid:12)(cid:12) ≥ nqj c ( nq ) = c nj . Similarly we show that | sin( npπ/α j ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin nq √ πpj + √ q !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ c nj . j ∈ N ∞ X n =1 | F ( n, j ) | β np − ≤ sup j ∈ N X n ∈ G j Cβ np − + c X n ∈ N \ G j njβ np − < ∞ . Now let ε > 0. First we note that b j = X n ∈ N \ G j njβ np − → , if j → ∞ . Hence there is j ∈ N such that for j > j we have b j < ε/ 3. We take n ∈ N such that P ∞ n = n nβ np − < ε/ (3 cj ) and P ∞ n = n β np − < ε/ (3 C ). Now let δ = β n p − . If µ ( G ) < δ then G ⊂ { n , n + 1 , . . . } andsup j ∈ N X n ∈ G | F ( n, j ) | β np − ≤ sup j ∈ N X n ∈ G ∩ G j Cβ np − + sup j ∈ N c X n ∈ G \ G j njβ np − ≤ C X n ∈ G β np − + c sup j>j X n ∈ N \ G j njβ np − + c X n ∈ G nj β np − ≤ ε ε ε ε . In the same way we prove uniform integrability of F ( n, j ) and F ( n, j ) andwe obtain the assertion of the lemma. Acknowledgements We thank Zbigniew Palmowski for introducing us into the subject. Wealso thank Michel Waldschmidt for discussions about this paper. References [1] A. Baker. A concise introduction to the theory of numbers . CambridgeUniversity Press, Cambridge, 1984.[2] V. Bernyk, R. C. Dalang, and G. Peskir. The law of the supremum of astable L´evy process with no negative jumps. Ann. Probab. , 36(5):1777–1789, 2008. 153] J. Bertoin. L´evy processes , volume 121 of Cambridge Tracts in Mathe-matics . Cambridge University Press, Cambridge, 1996.[4] N. H. Bingham. Maxima of sums of random variables and supremaof stable processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete ,26:273–296, 1973.[5] F. Caravenna and L. Chaumont. Invariance principles for random walksconditioned to stay positive. Ann. Inst. Henri Poincar´e Probab. Stat. ,44(1):170–190, 2008.[6] L. Chaumont, A. E. Kyprianou, and J. C. Pardo. Some explicit iden-tities associated with positive self-similar Markov processes. StochasticProcess. Appl. , 119(3):980–1000, 2009.[7] D. A. Darling. The maximum of sums of stable random variables. Trans.Amer. Math. Soc. , 83:164–169, 1956.[8] R. A. Doney. On Wiener-Hopf factorisation and the distribution ofextrema for certain stable processes. Ann. Probab. , 15(4):1352–1362,1987.[9] P. Graczyk and T. Jakubowski. On exit time of symmetric α -stableprocesses. Preprint, 2009[10] I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series, and prod-ucts . Elsevier/Academic Press, Amsterdam, seventh edition, 2007.[11] A. Kuznetsov. Wiener-Hopf factorization and distribution of extremafor a family of L´evy processes. To appear in J.Applied Prob., 2009.[12] A. E. Kyprianou. Introductory lectures on fluctuations of L´evy processeswith applications . Universitext. Springer-Verlag, Berlin, 2006.[13] A. E. Kyprianou and Z. Palmowski. Fluctuations of spectrally negativeMarkov additive processes. In S´eminaire de probabilit´es XLI , volume1934 of