Wavelet Coefficients of Levy Process Ruslan Suyundykov, Stéphane Puechmorel, Louis Ferré

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Ruslan Suyundykov, Stéphane Puechmorel, Louis Ferré. Wavelet Coefficients of Levy Process. 42èmes Journées de Statistique, 2010, Marseille, France, France. ￿inria-00494703￿

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R. Suyundykov S. Puechmorel L. Ferre ENAC Dept. MI LMA ENAC Dept. MI LMA Universite Toulouse Le Mirail 7, Avenue Edouard Belin 7, Avenue Edouard Belin 5, alles Antonio Machado 31055 Toulouse FRANCE 31055 Toulouse FRANCE 31058 Toulouse FRANCE email: [email protected] email: [email protected] email: [email protected]

Abstract—Cet article presente une expression de la fonction caracteristique des coefficients de decomposition en ondelettes d’un processus de Levy. Le cas particulier de l’ondelette de Haar et d’un processus entrelace est traite plus en detail. Abstract—The main object of the paper is to study the wavelet decomposition of Levy processes by wavelets with compact support. The general result was applied to the interlacing process with finite different jumps by Haar wavelets. alt Keywords : Ondelettes, Processus

I.INTRODUCTION Estimation of wavelet coefficients of Levy process is a research subject of both theoretical and practical interest as well as its use in a large number of practical application.

The statement of problem came from exploring of aircraft 0 10000 20000 30000 trajectories. Fig. 1 and Fig. 2 provide a prototype for the type of altitude of aircraft landing trajectories and vertical velocity. −2500 −2000 −1500 −1000 −500 0 Exploring these different trajectories we can conclude that time the velocity of the altitude is an interlacing process. Wavelet decomposition of the altitude velocity involved us to get more general result, i.e. wavelet composition of Levy process by wavelets with compact support. Figure 1. The altitude of 10 aircraft trajectories. The theory of stochastic processes was one of the important mathematical developments of the twentieth century. The tools with which this is made precise were provided by A. Levy: he made the connection with infinitely divisible distri- N. Kolmogorov in the 1930s. In this article we will study butions (The Levy-Khintchine Formulae) and described their wavelet analysis of the class of stochastic processes called structure. Let us give a definition of Levy process: Levy processes. Paul Levy first studied them in the 1930s. Many scientists worked on getting wavelet coefficients of Definition 1. A Levy process X = (Xt)t≥0 = (X(t, ω))t≥0 Levy process particular case and achieved definite results is a stochastic process satisfying the following: [5], [6]. J.C. Simon de Miranda worked on probability density function of the empirical wavelet coefficients of a Poisson 1) Each X0 = 0 (with probability one), process. J. Istas calculated wavelet coefficients of a Gaussian process. The idea of this article is to find wavelet coefficients 2) X has independent and stationary increments, of general form of the Levy process. 3) X is stochastically continuous, i.e. for all ε > 0 and II.SOME BASICS AND NOTATIONS for all s ≥ 0

A. Levy process lim P(|Xt − Xs| > ε) = 0. Levy processes, i.e. processes in continuous time with t→s stationary and independent increments, are named after Paul Here and in the future we will suppose that we have are called Levy measures.

B. Wavelets For many years, the sine, cosine and imaginary exponential functions have been the basic functions of analysis. The se- quence (2π)1/2eikx, k = 0, ±1, ±2,... forms an orthonormal basis of the standard space L2[0, 2π]; Fourier series are the P ikx linear combinations ake . Their study has been and remains, an unquenchable source of problems and discoveries in mathematical analysis. The problem arise from the absence velocity of a good dictionary for translating the properties of a function into those of its Fourier coefficients. At the beginning of 1980s, many scientists were already us- ing ”wavelets” as an alternative to traditional Fourier analysis. This alternative gave grounds for hoping for simpler numerical analysis and more robust synthesis of certain transitory phe-

−50 −40 −30 −20 −10 0 nomena. The theory of wavelets was developed by Y.Meyer, I.Daubechies, S.Mallat and others in the end of 1980s (see [1], 0 500 1000 1500 [3], [4]). Index The word ”wavelet” is used in mathematics to denote a kind of orthonormal bases in L2 with remarkable approximation properties. This means that any f ∈ L2(R) can be represented as a series (convergent in L2(R)): Figure 2. The velocity of aircraft trajectory altitude. X X∞ X f(x) = αkϕ0k(x) + βjkψjk(x) (2.1) 0 0 Levy process modification X = (X (t))t≥0 of process k j=0 k 0 0 X = (X(t))t≥0, such that X = X (t, ω) is right-continuous with left limit, see [4]. where αk, βjk are some coefficients, and ψjk, k ∈ Z, is a To understand the structure of a generic Levy process, basis for Wj which is called resolution level of multiresolution we employ Fourier analysis. The characteristic functions of analysis. The wavelet expansion needs to justify the use of Levy process were completely characterized by Levy and j/2 j ψjk(x) = 2 ψ(2 x − k) (2.2) Khintchine in the 1930s. in (2.1), i.e. the existence of such a function called mother Theorem 1. The Levy-Khintchine Formula. If X = wavelet.

tη(θ) {X(t)}t≥0 is a Levy process, then φt(θ) = e , for each III.WAVELET COMPOSITION OF LEVYPROCESSBY t ≥ 0, θ ∈ R, where WAVELETS WITH COMPACT SUPPORT 1 η(θ) = iaθ − σ2θ2+ Let X be a Levy process satisfying E[X(1)] = 2 Z 2 ¡ ¢ a0,E[X(1) ] < ∞. Let ψ(t) be a mother wavelet with iθy + e − 1 − iθyI||y||<1(y) ν(dy), compact support, such that supp ψ(t) ∈ R . We can define a R/{0} function for some a ∈ R, σ ≥ 0 and a Borel measure ν on R/{0} for Z+∞ ξ : ω 7→ ξ(ω) = X(t, ω)ψ(t)dt, (3.1) which Z −∞ 2 (|y| ∧ 1)ν(dy) < ∞. where we assume, that X(t, ω) = 0, ∀t < 0. The existence R/{0} of that integral follows from existence of cadlag Levy process Conversely, given a mapping of the form (1) we can always modification.

tη(θ) construct a Levy process for which φt(θ) = e . Theorem 2. The function ξ is a random variable, such that The triple (a, σ2, ν) is called the characteristics of X. It R E[ξ] = a0 tψ(t)dt. determines the law pt . The measures ν that can appear in (1) R Proof. Let supp ψ(t) ⊂ [0, b], T = {0 = x0 = t0 < t1 < equation ··· < tn = xn = b} be a target partition. We can define a random process Z+∞

ξij (ω) = X(t, ω)ψij (t)dt = Xn ξ = X(t , ω)ψ(t )∆x . (3.2) −∞ T i i i Z+∞ i=1 j X(t + , ω)ψ (t)dt = 2i i0 Obviously, ∀ ω ∈ Ω ⇒ lim ξT (ω) = ξ(ω). It means, that ξ −∞ ∆→0 Z+∞µ ¶ is measurable function or random variable. j j X(t + , ω) − X( , ω) ψ (t)dt ∼ Then 2i 2i i0 −∞   Z Z Z Z+∞ Eξ = ξ(ω)dP =  X(t, ω)ψ(t)dt dP = X(t, ω)ψi0(t)dt. −∞ Ω  Ω R  Z Z IV. THEINTERLACINGPROCESSCOMPOSITIONBY HAAR  X(t, ω)ψ(t)dP dt = R Ω  WAVELET Z Z Z Let C (t) = at + B (t) be a Brownian motion with ψ(t)  X(t, ω)dP dt = a0 tψ(t)dt. a,σ σ NPλ(t) R Ω R drift. Let Zλ(t) = Yj be a Compound Poisson process, j=1 where (Y , n ∈ N) is a sequence of independent identically Let us calculate the characteristic function of the random n distributed random variables with common law q and N is variable ξ: λ an independent Poisson process. +R∞ We can define a Levy process by the prescription X(t) = Theorem 3. Let Ψ(t) = ψ(s)ds. The characteristic C (t) + Z (t), provided the two summands are assumed t a,σ λ function of ξ is φ(θ) = eη(θ), where to be independent. We call this an interlacing process since its paths have the form of continuous motion interlaced with Z Z random jumps of size ||Yn|| occurring at the random times τn. 1 2 2 2 X has characteristic exponent η(θ) = ia0θ Ψ(t)dt − σ θ Ψ(t) dt+ 2 Z R R 1 Z Z η(θ) = iaθ − σ2θ2 + (eiθy − 1)λq(dy), (4.1) iθΨ(t)x 2 (e − 1 − iθΨ(t)x) dt ν(dx) R R R which is quite close to the general form in theorem. Indeed was proposed as the form of the most general η by the Italian Proof of the theorem 3 see in the appendix. mathematician Bruno de Finetti in the 1920s. His error was in Let ψ (x) = 2i/2ψ(2ix − j) be a wavelet, where ψ(x) a ij failing to appreciate that the finite measure λq can be replaced mother wavelet. Let us define ξ as: ij by a σ-finite Levy measure ν. But if we do this, (eiθy −1) may not be ν-integrable and hence we must adjust the integrand. +∞ Z Let us see a simple example: the Interlacing Process ξ = ξ (ω) = X(t, ω)ψ (t)dt. (3.3) ij ij ij (Xt)t≥0 composition by Haar wavelet H(t), where Yi dis- n −∞ crete random variables with possible values {ai}i=1 and the probability P(Y = ai) = pi. Theorem 4. In our conditions Definition 2. The Haar wavelet is the function defined on the 1) ξij are identically distributed random variables ∀j ∈ Z real line R as and fixed i;  T  1  −1, t ∈ [− 2 , 0]; 2) if ∀j1, j2 that supp ψij1 supp ψij2 = ∅ then ξij1 , ξij2  H(t) = 1, t ∈ (0, 1 ]; (4.2) are independent.  2   0, otherwise. Proof. These follow from definition of Levy process and Therefore, where  Z Z+∞ 1 1 1 2 2 iθx  2 + t, t ∈ [− 2 , 0]; η(θ) = ia0θ − σ θ + (e − 1 − iθx)ν(dx), Ψ(t) = H(s)ds = 1 − t, t ∈ (0, 1 ]; (4.3) 2  2 2 R 0, otherwise. R t 2 and a0 = E[X1], R(|x | ∧ |x|)ν(dx) < ∞. and Let us use the same notations as in theorem 2. Then

Z Z Pn 1 2 2 2 ibiθ(X(ti)−X(ti−1)) η (θ) = ia θ Ψ(t)dt − σ θ Ψ(t) dt+ iθξT i=1 ξ 0 2 Ee = Ee = Z Z R R Yn Yn Eeibiθ(X(ti)−X(ti−1)) = e(ti−ti−1)η(biθ) = (eiθΨ(t)x − 1 − iθΨ(t)x)dtν(dx) = i=1 i=1 R R Pn ³ ´ µ ¶ (ti−ti−1)η(biθ) a 1 σ2 ei=1 , i 0 θ − θ2+ 4 2 12 where Xn Z Xn iθΨ(t)ai λ pi (e − 1)dt = bi = ψ(tj)∆xj. i=1 j=i R µ ¶ ³a ´ 1 σ2 We have the next: i 0 θ − θ2+ 4 2 12 Xn Z Ã µ ¶! Xn (ti − ti−1)η(biθ) = iθy y λ (e − 1)d piFsign(ai) , i=1 |ai| n n R i=1 X 1 X ia θ (t − t )b − σ2θ2 (t − t )b2+ 0 i i−1 i 2 i i−1 i where  i=1 i=1  0, s < 0; Xn Z F (s) = 2s, s ∈ [0, 1 ]; (t − t ) (eibiθx − 1 − ib θx)ν(dx). +  2 i i−1 i 1 i=1 1, s > 2 . R and  Let us to calculate the 1 n Z  0, s < − 2 ; X 1 ibiθx lim (ti − ti−1) (e − 1 − ibiθx)ν(dx). F−(s) = 1 + 2s, s ∈ [− 2 , 0];  ∆T →0 1, s > 0. i=1 R It means that ξ has the same distribution as C √ (1)+ ∀ε > 0 ∃δ1 = δ1(ε) > 0 : ∀T ∆T < δ1 ⇒ a/4,σ/ 12 ¯ ¯ Nλ(1) ¯ Z+∞ ¯ P ¯ Xn ¯ Ti where Ti have common distribution function ¯ ψ(s)ds − ψ(t )∆x ¯ < ε. i=1 ¯ i i¯ Pn ¯ j=i ¯ s xi piF ( ). sign(ak) |ak| i=1 Let θ be fixed. Then ∀ε1 > 0 ∃Nε1 > 0 : Z V. CONCLUSION (2 + C1θ|x|)ν(dx) < ε1, We determined in this paper the general expression of the R\[−Nε1 ,Nε1 ] characteristic function and therefore probability distribution where C1 = max(ψ(t))b. function of wavelet coefficients of Levy process. We con- t∈R sidered the special case of the Haar wavelet and interlacing And it is simple to get, that process for which we calculated probability distribution func- Xn tion. |bi| ≤ max(ψ(t)) ∆xi ≤ C1. t∈R In the future works estimation of wavelet coefficients of j=i Levy processes will be implemented in different fields such Therefore, as the case of aircraft landing trajectories, financial data, etc. ¯ ¯ ¯ ¯ ¯Xn Z ¯ ¯ iθbix ¯ ¯ (ti − ti−1) (e − 1 − iθbix)ν(dx)¯ ≤ VI.APPENDIX ¯ ¯ ¯i=1 ¯ R\[−Nε1 ,Nε1 ] Proof of Theorem 3: Xn Z As E[|X(t)|] < ∞, then (Levy-Khinchine formulae) (ti − ti−1) (2 + C1θ|x|)ν(dx) < ε1b. i=1 iθXt tη(θ) R\[−N ,N ] Ee = e , ε1 ε1 And Therefore, ¯ ¯ n Z ¯ Nε1 X ¯ n Z ¯ iθbix X ¯ lim (ti − ti−1) (e − 1 − iθbix)ν(dx) ¯ iθbix ¯ (ti − ti−1) (e − 1 − iθbix)ν(dx) ¯ ∆T →0 ¯ i=1 R i=1 −Nε N ¯ 1 Z ε1 Zb ¯ Nε ¯ ¯ n Z 1 ¯ − (eiθΨ(t)x − 1 − iθΨ(t)x)dtν(dx) < X ¯ ¯ − (t − t ) (eiθΨ(xi)x − 1 − iθΨ(x )x)ν(dx) = ¯ i i−1 i ¯ 0 ¯ −Nε1 i=1 −Nε ¯ 1 ¯ ε1b, ¯ Nε1 ¯ ¯Xn Z X+∞ j ¯ ¯ (iθx) j j ¯ and ¯ (ti − ti−1) (bi − Ψ(xi) )ν(dx)¯ ≤ ¯ j! ¯ n Z ¯i=1 j=2 ¯ X −Nε iθbix 1 lim (ti − ti−1) (e − 1 − iθbix)ν(dx) = N ∆T →0 Z ε1 i=1 Xn X+∞ j R (|θx|) j Z Z (ti − ti−1) ε(2C1) ν(dx) = j! (eiθΨ(t)x − 1 − iθΨ(t)x)dtν(dx). i=1 j=2 −Nε1 R R N Z ε1 It is similar to get: 2C1|θx| εb (e − 1 − 2C1|θx|)ν(dx). Xn Xn −Nε1 1 2 2 2 lim (ia0θ (ti − ti−1)bi − σ θ (ti − ti−1)bi ) = ∆T →0 2 i=1 i=1 Then, ∃δ2 = δ2(ε) > 0, that ∀T : ∆T < δ2 ⇒ Z Z 1 2 2 2 ia0θ Ψ(t)dt − σ θ Ψ(t) dt. ¯ 2 ¯ Zb ¯ R R ¯ (eiθΨ(t)x − 1 − iθΨ(t)x)dt ¯ Therefore, 0 ¯ Z Z Xn ¯ 1 2 2 2 ¯ η(θ) = ia0θ Ψ(t)dt − σ θ Ψ(t) dt+ − (t − t )(eiθΨ(xi)x − 1 − iθΨ(x )x)¯ < 2 i i−1 i ¯ i=1 R Z Z R ε(|x2| ∧ |x|) (eiθΨ(t)x − 1 − iθΨ(t)x) dt ν(dx).

R R ∀x ∈ [−Nε1 ,Nε1 ]. Therefore, ¥

¯ Nε REFERENCES ¯ Xn Z 1 ¯ iθΨ(xi)x ¯ (ti − ti−1) (e − 1 − iθΨ(xi)x)ν(dx) ¯ [1] I. Daubechies, The wavelet transform, time-frequency localization and i=1 −Nε 1 signal analysis, IEEE Trans. Inf. Th., 1990. N   ¯ Z ε1 Zb ¯ ¯ [2] W. Hardle, D. Picard, G. Kerkyacharian and A. Tsybakov, Wavelets, −  (eiθΨ(t)x − 1 − iθΨ(t)x)dt ν(dx)¯ < ¯ Approximation and Statistical Applications, Un premier resultat du −Nε 0 1 seminaire Paris-Berlin, 1997. N Z ε1 [3] Y. Meyer, Wavelets: Algorithms and Applications, SIAM, 1993. ε (|x2| ∧ |x|) ν(dx). [4] J. Jacod, A.N. Shiryaev, Limit Theorems for Stochastic Processes, −Nε 1 Springer series, 2002. Thus, [5] J. Istas, Wavelet coefficients of a Gaussian process and applications, ¯ Annales de l’I.H.P., section B, tome 28, No. 4, 1992, 537-556. ¯ n Z ¯ X ¯ (t − t ) (eiθbix − 1 − iθb x)ν(dx) [6] J.C. Simon de Miranda, Probability Density Functions of the Emper- ¯ i i−1 i i=1 R ical Wavelet Coefficients of a Wavelet Estimator of Multidimensional N ¯ Z ε1 Zb ¯ Poisson Intensities, IEEE, 2008. ¯ − (eiθΨ(t)x − 1 − iθΨ(t)x)dtν(dx)¯ < [7] A. Bijaoui, G. Jammal On the distribution of the wavelet coefficients ¯

−Nε1 0 for a Poisson noise, Signal Processing, 2000.

ε1b + εC(ε1).