A Fourier Approach for the Level Crossings of Shot Noise Processes with Jumps Hermine Biermé, Agnès Desolneux
To cite this version:
Hermine Biermé, Agnès Desolneux. A Fourier Approach for the Level Crossings of Shot Noise Processes with Jumps. Journal of Applied Probability, Cambridge University press, 2012, 49 (1), pp.100–113. hal-00537798v2
HAL Id: hal-00537798 https://hal.archives-ouvertes.fr/hal-00537798v2 Submitted on 14 Jul 2011
HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A FOURIER APPROACH FOR THE LEVEL CROSSINGS OF SHOT NOISE PROCESSES WITH JUMPS
HERMINE BIERME´ AND AGNES` DESOLNEUX
Abstract. We use here a change of variables formula in the framework of functions of bounded variation to derive an explicit formula for the Fourier transform of the level crossings function of shot noise processes with jumps. We illustrate the result on some examples and give some applications. In particular, it allows us to study the asymptotic behavior of the mean number of level crossings as the intensity of the Poisson point process of the shot noise process goes to infinity.
In this paper, we will consider a shot noise process which is a real-valued random process given by (1) X(t)= β g(t τ ), t R, i − i ∈ i Z X∈ where g is a given (deterministic) measurable function (it will be called the kernel function of the shot noise), the τi i Z are the points of a homogeneous Poisson point process on the line of intensity λ> 0, { } ∈ and the βi i Z are independent copies of a random variable β (called the impulse), independent of { } ∈ τi i Z. { Such} ∈ a process has many applications (see [16] and the references therein for instance) and it is a well-known and studied mathematical model (see [10], [15], [5] for some of its properties). We will be interested here in the level crossings of such a process. Usually, the mean number of level crossings of a stochastic process is computed thanks to a Rice’s formula (see [12] or [1]) that requires some regularity conditions on the joint probability density of X and of its derivative. This joint probability density is generally not easy to obtain and to study. Its existence is also sometimes a question in itself. This is why, instead of working directly on the mean number of level crossings, we will work on the Fourier transform of the function that maps each level α to the mean number of crossings of the level α per unit length. Thanks to a change of variables formula, we will be able to relate this Fourier transform to the characteristic function of the shot noise process (which, unlike the probability density, always exists and is explicit).
1. General result In [3], we have studied the level crossings of the shot noise process X when the kernel function g is smooth on R. We will consider here the case where g is a piecewise smooth function, that is not necessarily continuous. We first introduce some definitions and notations. Let I be an open interval of R and k 0 be an integer. A function f : I R is said piecewise Ck on I if there exists a finite set of points≥ of discontinuity of f on I, denoted→ by S = s , s ,...,s , with m 1 and s <...s t s,t
(3) and ∆f(s) := f(s+) f(s ). − − (k) Finally, when k 1, we will denote by f ′,...,f the functions that are defined at all points s / Sf ≥ (k) ∈ by the usual derivatives f ′(s),...,f (s). In the sequel, we will need assumptions on the kernel function g and on the impulse β of the shot noise process X defined by (1). These assumptions are grouped together into the following condition denoted by (C): 2 1 (C) : E( β ) < , g is piecewise C on R and g, g′, g′′ L (R). | | ∞ ∈ We will also denote the jump set of g on R by S = t ,t ,...,t , with t <... nP n T V (g, R) = sup g(ak) g(ak 1) = g′(s) ds + ∆g(tj) < , P | − − | R | | | | ∞ Xk=1 Z Xj=1 where the supremum is taken over all partitions P = a ,...,a of R with n 1 and a <...< { 0 nP } P ≥ 0 anP . Finally, we assume that the points τ of the Poisson point process are indexed by Z in such a way { i} that for any k N, one has 0 < τ0 < τ1 < τ2 < . . . < τk <... and 0 > τ 1 > τ 2 > . . . > τ k >.... ∈ − − − 1.1. Piecewise regularity of the shot noise process. The shot noise process “inherits” the regularity of the kernel function g. More precisely, we have the following result. Theorem 1. Assume that the condition (C) holds, then the shot noise process X defined by (1) is a stationary process which is almost surely piecewise C1 on any interval (a, b) of R. The jump set of X on (a, b) is S (a, b) where S = (τ + S ) X ∩ X i g i Z [∈ and t / S , X′(t)= β g′(t τ ). ∀ ∈ X i − i i Z X∈ Proof. Note that, since E( β ) < and g L1(R), for any t R, the random variable X(t) is | | ∞ ∈ ∈ well-defined and E(X(t)) = λE(β) R g(s)ds, according to [16]. Moreover, since the Poisson point process is homogeneous, X is a stationary process. R Let us first remark that since the jump set Sg of g contains exactly n points, we can write g as the sum of 2n piecewise C2 functions on R, each of them having only one discontinuity point and having the same regularity properties as g. Therefore we may and will assume that g has only one Z discontinuity point and we write Sg = t1 . Let i0 . We set Ii0 := [t1 + τi0 ,t1 + τi0+1], Then, for any t R, { } ∈ ∈ X(t)= β g(t τ )+ β g(t τ )+ β g(t τ )+ β g(t τ ). i − i i − i i0 − i0 i0+1 − i0+1 i>iX0+1 i g′(t s τi0 )= g′′(u s τi0 )du + g′(t1 s), − − t +τ − − − Z 1 i0 A FOURIER APPROACH FOR THE LEVEL CROSSINGS OF SHOT NOISE PROCESSES WITH JUMPS 3 such that by Fubini-Tonelli Theorem, 0 sup g′(t s τi0 ) ds (τi0+1 τi0 ) g′′(s) ds + g′(s) ds. t Ii | − − | ≤ − R | | R | | Z−∞ ∈ 0 Z Z Then 1 E βi sup g′(t τi) λE( β ) g′′(s) ds + g′(s) ds < + , | | | − | ≤ | | λ R | | R | | ∞ à t Ii0 ! i b f ∗(s) (4) h(α)Nf (α, (a, b)) dα = h(f(s)) f ′(s) ds + h(α) dα. R a | | f (s) Z Z s Sf (a,b) Z ∗ ∈ X∩ 4 HERMINE BIERME´ AND AGNES` DESOLNEUX Proof. Let us assume that S (a, b)= s ; 1 j m with m 1 and s := a < s <... sj+1 h(α)# s (sj, sj+1) ; f(s)= α dα = h(f(s)) f ′(s) ds. R { ∈ } | | Z Zsj Then, we sum up these equalities for j = 0 to m, and using (5), this achieves the proof of the proposition. ¤ Notice that a change of variables formula for functions of bounded variation was also obtained in [6]. But their formula is valid away from the jump set of f, whereas our formula (4) takes explicitly into account the contribution of the jump set. Let us also mention the fact our formula (4) is a kind of extension of Banach’s Theorem used by Rychlik in [17] (p.335). The main difference is that our formula involves a function h that is not necessarily constant, but on the other hand, we are restricted to a piecewise C1 function f (whereas Banach’s Theorem is valid for functions of bounded variation). Let us be now interested in the shot noise process X. For α R, let NX (α) be the random variable that counts the number of crossings of the level α by the process∈ X in the interval (0, 1). Using (2), it is defined by NX (α) = # t (0, 1) ; X (t) α X∗(t) . { ∈ ∗ ≤ ≤ } We will be mainly interested in its expectation, namely in CX (α)= E(NX (α)). The function α CX (α) is called the mean level crossings function and we compute its Fourier transform in the7→ following theorem. This result has to be related to the heuristic approach of [2] (in particular their formula (13) that involves the joint density of X(t) and X′(t) in a Rice’s formula - but without checking any of the hypotheses for its validity). Our theorem will involve the characteristic function of the shot noise process which is easily computable. Actually, if we denote for all u,v R, ∈ (6) ψ(u,v)= E(eiuX(0)+ivX′(0)) and F (u)= E(eiuβ) then it is well-known (see [16] for instance) that b (7) ψ(u,v) = exp λ (F (ug(s)+ vg′(s)) 1) ds . R − µ Z ¶ Theorem 2. Assume that the condition (C) isb satisfied, then the mean level crossings function CX belongs to L1(R) and CX (α) dα = E(T V (X, (0, 1))) λE( β )T V (g, R). R ≤ | | Z Moreover, its Fourier transform, denoted by u C (u) is given for u = 0 by 7→ X 6 n 1 iuX(0) iuX(0) iu(βg)∗(tj ) iu(βg) (tj ) C (u)= E(e X′(0) )+ λE(e )d E(e ) E(e ∗ ) , X | | · iu − Xj=1 ³ ´ and for ud= 0 by n C (0) = E(T V (X, (0, 1))) = E( X′(0) )+ λE( β ) ∆g(t ) . X | | | | | j | Xj=1 d A FOURIER APPROACH FOR THE LEVEL CROSSINGS OF SHOT NOISE PROCESSES WITH JUMPS 5 Proof. According to Theorem 1, we can apply Proposition 1 such that, for any bounded continuous functions h defined on R, almost surely 1 X∗(t) (8) h(α)NX (α) dα = h(X(t)) X′(t) dt + h(α) dα. R 0 | | X (t) Z Z t SX (0,1) Z ∗ ∈ X∩ Taking h = 1, we obtain that 1 NX (α) dα = X′(t) dt + ∆X(t) = T V (X, (0, 1)). R 0 | | | | Z Z t SX (0,1) ∈ X∩ Using the stationarity of X , we have E 1 X (t) dt = E( X (0) ) λE( β ) g (s) ds and ′ 0 | ′ | | ′ | ≤ | | R | ′ | ³ ´ R n R E ∆X(t) = λE( β ) ∆g(t ) . | | | | | j | t SX (0,1) j=1 ∈ X∩ X Therefore, n CX (α) dα λE( β ) g′(s) ds + ∆g(tj) λE( β )T V (g, R). R ≤ | | R | | | | ≤ | | Z Z Xj=1 Now, taking h(α)= eiuα for some u R with u = 0 in (8), we get ∈ 6 1 X∗(t) iuα iuX(t) iuα e NX (α) dα = e X′(t) dt + e dα, R 0 | | X (t) Z Z t SX (0,1) Z ∗ ∈ X∩ 1 n iuX(t) (9) = e X′(t) dt + A(j), 0 | | Z Xj=1 1 iuX((tj +τi)+) iu max(βi∆g(tj ),0) iu min(βi∆g(tj ),0) where A(j) := iu τi ( tj ,1 tj ) e e e . Now, let us write ∈ − − − X((t + τ )+) = β g(t + τ τ )+ β g(t +) and j i P k j i − k i ¡ j ¢ τk=τi X6 1 iu(βig)∗(tj ) iu(βig) (tj ) B(u,t , β ) := e e ∗ , j i iu − ³ ´ such that A(j)= exp iu β g(t + τ τ ) B(u,t , β ). k j i − k j i τi ( tj ,1 tj ) k=i ∈ −X − X6 Then, for M > max1 j n tj + 1, we consider ≤ ≤ | | AM (j)= exp iu βkg(tj + τi τk) B(u,tj, βi), such that AM (j) A(j) a.s.. − M−→+ τi ( tj ,1 tj ) k=i; τk M → ∞ ∈ −X − 6 X| |≤ We have that NM NM d AM (j) = 1( tj ,1 tj )(Ui)exp iu βkg(tj + Ui Uk) B(u,tj, βi), − − − i=1 k=1,k=i X X6 where (Uk)k N is an i.i.d. sequence of random variables of uniform law on [ M,M] independent ∈ − from (βk)k N, and NM is a Poisson random variable of parameter 2λM, independent from (Uk)k N ∈ ∈ 6 HERMINE BIERME´ AND AGNES` DESOLNEUX 0 and from (βk)k N. We adopt the convention that = 0. Now, by conditioning, we get ∈ Xi=1 + ∞ E(A (j)) = E(A (j) N = m)P[N = m], M M | M M m=0 X and then computing E(AM (j) NM = m), for all m 0, using the independence of (Uk)k N and | ≥ ∈ (βk)k N, finally leads to ∈ 1 tj M+x+tj − E(AM (j)) = λE(B(u,tj, β)) exp λ F (ug(s)) 1 ds dx, tj à M+x+tj − ! Z− Z− ³ ´ where F is given by (6). As a consequence, using Lebesgue’s Theoremb and Formula (7) for E(eiuX(0)), we get iuX(0) b E(A(j)) = λE(e )E(B(u,tj, β)). Finally, taking the expectation on both sides of Equation (9) and using the stationarity of X, leads to the announced result for CX (u). ¤ Proposition 2. Under the condition (C), if we assume moreover that E(β2) < and that g d ′ L2(R), then, for u = 0, ∞ ∈ (10) 6 n 1 + 1 ∂ψ ∂ψ 1 ∞ iu(βg)∗(tj ) iu(βg) (tj ) CX (u)= − (u,v) (u, v) dv +λψ(u, 0) E(e ) E(e ∗ ) , π 0 v ∂v − ∂v − iu − Z µ ¶ Xj=1 ³ ´ whered ψ is given by (6). 1 Proof. Since g′ L (R) and since g has a finite number of discontinuity points with finite left and ∈ 1 2 right limits, it follows that g L∞(R). Consequently, g L∞(R) L (R) L (R). Therefore, 2 ∈2 ∈ ∩ ⊂ 2 2 when E(β ) < and g′ L (R), the characteristic function ψ of (X(0), X′(0)) is C on R . + 1 ∂ψ∞ ∂ψ∈ Then, ∞ (u,v) (u, v) dv is well-defined and is the Hilbert transform of the function 0 v ∂v − ∂v − ∂ψ v R(u,v).³ Moreover, the computations´ of Theorem 1 in our previous paper [3] yield 7→ ∂v + iuX(0) 1 ∞ 1 ∂ψ ∂ψ E(e X′(0) )= − (u,v) (u, v) dv. | | π v ∂v − ∂v − Z0 µ ¶ This ends the proof of the proposition. ¤ 2. A particular case The formula for CX (u) can become simpler in some cases. The first particular case is the one when iuX(0) the kernel g is piecewise constant, since then X′(0) = 0 a.s. and thus the term E(e X′(0) ) vanishes. Let us alsod make the following remark. | | Remark 2. When β 0 a.s., then for any h < h R, the function u 1 ψ(u, 0) (F (uh ) ≥ 1 2 ∈ 7→ iu · 2 − F (uh1)) is the Fourier transform of the function α P[α βh2 X(0) α βh1], where β and X(0) are taken independent. 7→ − ≤ ≤ − b b We will now give a simpler formula for CX (u) in the case where g is piecewise non-increasing (meaning that g 0 and thus g is non-increasing on each of the intervals on which it is continuous, ′ ≤ but it can have jumps tj such that ∆g(tj)= gd(tj+) g(tj ) > 0). In that case, we have the following proposition. − − Proposition 3. Assume that condition (C) holds. Assume moreover that β 0 a.s. and that g′ 0. Then, for all u R, ≥ ≤ ∈ 1 (11) C (u) = 2λψ(u, 0) F (ug(t +)) F (ug(t )) , X · iu j − j− tj :∆Xg(tj )>0 ³ ´ d b b A FOURIER APPROACH FOR THE LEVEL CROSSINGS OF SHOT NOISE PROCESSES WITH JUMPS 7 where ψ and F are given by (6). As a consequence, for almost every α R, ∈ (12) C (α) = 2λ P[α βg(t +) X(0) α βg(t )], b X − j ≤ ≤ − j− tj :∆Xg(tj )>0 where β and X(0) are taken independent. Proof. Since g 0, we have X (0) 0 a.s. and consequently ′ ≤ ′ ≤ iuX(0) iuX(0) ∂ψ E(e X′(0) )= E(e X′(0)) = i (u, 0). | | − ∂v Now, since g is piecewise non-increasing and belongs to L1(R), we have lim g(s) = 0 and thus s + | |→ ∞ lim F (ug(s)) = 1. Then, recalling that ψ(u,v) = exp λ R(F (ug(s)+ vg′(s)) 1) ds , we get s + − | |→ ∞ ³ R ´ ∂ψ b b (u, 0) = λψ(u, 0) g′(s)F ′(ug(s)) ds ∂v R Z t1 n 1 tj + b − +1 ∞ = λψ(u, 0) g′(s)F ′(ug(s)) ds + g′(s)F ′(ug(s)) ds + g′(s)F ′(ug(s)) ds · tj tn Z−∞ Xj=1 Z Z b n 1 b b 1 − = λψ(u, 0) F (ug(t )) + (F (ug(t )) F (ug(t +))) F (ug(t +)) · u 1− j+1− − j − n Xj=1 n 1 b b b b = λψ(u, 0) (F (ug(t )) F (ug(t +))). · u j− − j Xj=1 Consequently, by Theorem 2, web get b n i C (u) = λψ(u, 0) (F (ug(t )) F (ug(t +))) X · u j− − j Xj=1 d 1 b b 1 +λψ(u, 0) F (ug(t +)) F (ug(t )) + F (ug(t )) F (ug(t +)) · iu j − j− iu j− − j tj :∆Xg(tj )>0 ³ ´ tj :∆Xg(tj )<0 ³ ´ 1 b b b b = 2λψ(u, 0) F (ug(t +)) F (ug(t )) . · iu j − j− tj :∆Xg(tj )>0 ³ ´ b b This achieves the proof of Formula (11) for CX (u). Formula (12) for CX (α) for almost every α R follows from Remark 2 and this ends the proof of the proposition. ∈ ¤ d A particular case of this proposition is when we make the additional assumption that the function g is positive and that it has only one positive jump at t1 = 0 from the value g(0 ) = 0 to the value g(0+) > 0. Formula (11) then simply becomes − 1 (13) C (u) = 2λψ(u, 0) (F (ug(0+)) 1). X · iu − This framework corresponds tod the case studied by Hsingb in [11], and where he considers up-crossings that are defined in the following way : the point t is an up-crossing of the level α by the process X if it is a point of discontinuity of X and if X(t ) α and X(t+) > α. Then, Hsing proves that the − ≤ (J) expected number of such points in (0, 1), denoted by UX (α), is given by (J) (14) U (α)= λ P[α βg(0+) < X(0) α], X − ≤ where β and X(0) are taken independent. Note that the left strict inequality comes from the way Hsing defines up-crossings of the level α. Now, let us consider up-crossings of the level α as usually defined (see [7] p.192): the point t is an up-crossing of the level α by the process X if there exists ε > 0 such that X(s) α in (t ε, t) and X(s) α in (t,t + ε); and let us denote by U (α) their ≤ − ≥ X 8 HERMINE BIERME´ AND AGNES` DESOLNEUX expected number in (0, 1). In a similar way we can define down-crossings of the level α. Let us remark that, since X is piecewise non-increasing, if moreover X is a.s. not identically equal to α in any interval of (0, 1) (this is satisfied for instance when P[X(0) = α] = 0), then the crossings of the level α are either up-crossings as defined by Hsing or down-crossings of the level α. Moreover, by stationarity of the process X, the expected number of down-crossings of the level α is equal to UX (α) and the result of Hsing yields to (15) C (α) = 2U (α) = 2λ P[α βg(0+) < X(0) α]. X X − ≤ The result of Hsing given by Equation (14) is stronger than the similar formula given by Equation (12) because his formula is valid for all levels α and moreover he needs weaker assumptions on the regularity of g. On the other hand, his proof strongly relies on the fact that g has only one jump and that g is identically 0 before that jump, and thus it can not be generalized to other kernel functions g. Finally, let us end this section by mentioning that we have studied here the case of a piecewise non- increasing kernel function g, but that, of course, similar results hold for a piecewise non-decreasing kernel. 3. High intensity and Gaussian limit We assume here that the assumptions of Proposition 2 hold. It is then well-known (see [15, 10] for instance) that, as the intensity λ of the Poisson point process goes to infinity, the normalized process Zλ defined by Xλ(t) E(Xλ(t)) t Zλ(t)= − , 7→ √λ where Xλ denotes a shot noise process (as defined by Equation (1)) with intensity λ for the ho- mogeneous Poisson point process, converges to a centered Gaussian process with covariance R(t) = 2 E(β ) R g(s)g(s t) ds. Now, how does− the number of level crossings of Z behave as λ goes to + ? To answer this, we R λ ∞ first determine the asymptotic expansion of the Fourier transform of CZλ as λ + . For u R, we have: → ∞ ∈ u E 1 u iuE(Xλ(t))/√λ 1 i (Xλ(0) (Xλ(0))) C (u) = C e− = E e √λ − X′ (0) Zλ √ Xλ √ √ | λ | λ µ λ¶ λ n ³ ´ u u u d dλ i (Xλ(0) E(Xλ(0))) i (βg)∗(tj ) i (βg) (tj ) + E e √λ − E(e √λ ) E(e √λ ∗ ) iu − ³ ´ Xj=1 ³ ´ As we have already studied it in Section 4 of [3], the first term of the right-hand side admits a limit as λ + , and more precisely, as λ + , → ∞ → ∞ 1 i u (X (0) E(X (0))) 2m2 2 E √λ λ λ m0u /2 e − Xλ′ (0) = e− + o(1), √λ | | π ³ ´ r 2 2 2 2 where m0 = E(β ) R g (s) ds and m2 = E(β ) R g′ (s) ds. For the second term, it is the product of two terms and each of them admits an asymptotic expansion as λ + . Indeed, assuming moreover that g LR3(R) and E( β 3) < , we haveR → ∞ ∈ | | ∞ u i (Xλ(0) E(Xλ(0))) u E e √λ − = exp λ (F ( g(s)) 1)ds iu√λE(β) g(s) ds R √λ − − R ³ ´ µ Z Z ¶ m u2 im u3 1 = exp 0 b 3 + o − 2 − √ √ µ 3 λ µ 붶 2 m0u /2 2ium3 1 = e− 1+ + o , √ √ µ 3m0 λ¶ µ λ¶ A FOURIER APPROACH FOR THE LEVEL CROSSINGS OF SHOT NOISE PROCESSES WITH JUMPS 9 where m = E(β3) g3(s) ds. And for a given jump number j, 1 j n, we have 3 R ≤ ≤ √ u u λ i (βg)∗R(tj ) i (βg) (tj ) iu 2 2 1 E(e √λ ) E(e √λ ∗ ) = E( β ) ∆g(tj) + E(β ) ∆g (tj) + o . iu − | | | | 2√λ | | √λ ³ ´ µ ¶ Finally, we thus have n 2 m0u /2 C (u) = √λ e− E( β ) ∆g(t ) Zλ | | | j | Xj=1 d n n 2 2m2 2ium3 iu 2 2 m0u /2 + + E( β ) ∆g(t ) + E(β ) ∆g (t ) e− + o(1). π 3m | | | j | 2 | j | r 0 j=1 j=1 X X Let us comment this result. When there are no jumps we obtain that CZλ (u) converges, as λ goes to 2 2 2m2 m0u /2 √m2 α /2m0 infinity, to e− . This implies that CZ (α) weakly converges to e− , which is the π λ dπ√m0 usual Rice’sq formula for the mean number of level crossings of Gaussian processes (see [7] p.194 for √ instance). Now, when there are jumps, the behavior of CZλ (u) is different, since the main term in λ 2 doesn’t vanish anymore. More precisely we have that 1 C (u) goes to e m0u /2E( β ) n ∆g(t ) , √λ Zλ − | | j=1 | j | which implies that d P n d 1 1 2 α /2m0 E CZλ (α) e− ( β ) ∆g(tj) in the sense of weak convergence. √ λ−→ √2πm | | | | λ →∞ 0 Xj=1 Notice also that taking u = 0 in the asymptotic formula for CZλ (u) gives the asymptotic behavior of the total variation of Zλ. Indeed, we then have n d 2m E(T V (Z , (0, 1))) = C (0) = √λ E( β ) ∆g(t ) + 2 + o(1). λ Zλ | | | j | π j=1 r X This kind of asymptotic behavior hasd also been studied by B. Galerne in [9] in the framework of random fields of bounded variation. 4. Some examples 4.1. Step functions. We start this section with some examples of explicit computations in the case of step functions. (1) The first simple example of step function is the one where the kernel g is a rectangular function : g(t) = 1 for t [0,a] with a> 0 and 0 otherwise. Notice that this is a very simple framework, that also fits∈ in the results of Hsing [11]. In this particular case X is piecewise constant almost surely. If the impulse β is such that β = 1 a.s. we will prove that • + k ∞ λa (λa) R N (16) CX (α)= 2λe− 1I k<α µ If the impulse β follows an exponential distribution of parameter µ> 0, then F (u)= µ iu • − iu and a simple computation gives ψ(u,v) = exp λa µ iu . On the one hand, the law of − k b 2aλ ³ ´+ aλ (2aλ) X(0) can be computed: PX(0)(dx) = e− δ0(dx)+ k=1∞ e− k! fµ,k(x)dx, where fµ,k is the probability density of the Gamma distribution of parameters µ and k, and δ0 is the Dirac measure at point 0. Then, P[X(0) = 0] > 0P such that C (0) = + and, for X ∞ all α > 0, P[X(0) = α] = 0 such that, according to (15), CX is continuous on (0, + ). On the other hand, ∞ iu 1 (17) C (u) = 2λ exp λa . X µ iu µ iu µ − ¶ − We recognize hered the Fourier transform of a non-central chi-square distribution, and thus aλ µα (18) for all α> 0, CX (α) = 2λµe− − I0(2 aλµα), where I0 is the modified Bessel function of the firstp kind of order 0; it is given by 1 π x cos θ + x2m I0(x)= π 0 e dθ = m∞=0 4m(m!)2 . (2) A second example is a “double rectangular” function given by: g(t) = 1if 1 t < 0; R P g(t)= 1 if 0 t< 1, and g(t) = 0 otherwise. Notice that this case does not fit− anymore≤ in the framework− ≤ of Hsing [11]. However X is still piecewise constant almost surely. If β = 1 almost surely, then simple computations show that • sin u C (u) = 4λ exp(2λ(cos u 1)). X u − The last term above is the characteristic function of the difference of two independent d Poisson random variables of same parameter λ (it is also called a Skellam distribution). Thus, as previously we obtain that C (α)=+ for all α Z and X ∞ ∈ + + k+2n R Z ∞ Z 2λ ∞ λ α , CX (α)= 4λpk1I k<α Acknowledgments This work has been supported by the French National Research Agency under the grant ANR-09- BLAN-0029-01. We also would like to thank the anonymous reviewer for his valuable comments and suggestions. References [1] Azais, J. and Wschebor, M. (2009). Level sets and extrema of random processes and fields. John Wiley And Sons Ltd, United Kingdom. [2] Bar-David, I. and Nemirovsky, A. (1972). Level crossings of nondifferentiable shot processes. IEEE Transactions on Information Theory 18, 27–34. [3] Bierme,´ H. and Desolneux, A. (2011). Crossings of smooth shot noise processes. Technical report No 2011-16. pr´epublication du MAP5. 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Extremes 3:4, 331–348. Hermine BIERME,´ MAP5 (UMR CNRS 8145) Universite´ Paris Descartes, 45 rue des Saints-Peres,` 75006 PARIS France E-mail address: [email protected] Agnes` Desolneux, MAP5 (UMR CNRS 8145) Universite´ Paris Descartes, 45 rue des Saints-Peres,` 75006 PARIS France E-mail address: [email protected]