Wavelet Compressibility of Compound Poisson Processes

Abstract

In this paper, we characterize the wavelet compressibility of compound Poisson processes. To that end, we expand a given compound Poisson process over the Haar wavelet basis and analyse its asymptotic approximation properties. By considering only the nonzero wavelet coefficients up to a given scale, what we call the sparse approximation, we exploit the extreme sparsity of the wavelet expansion that derives from the piecewiseconstant nature of compound Poisson processes. More precisely, we provide nearly-tight lower and upper bounds for the mean L2-sparse approximation error of compound Poisson processes. Using these bounds, we then prove that the sparse approximation error has a sub-exponential and super-polynomial asymptotic behavior. We illustrate these theoretical results with numerical simulations on compound Poisson processes. In particular, we highlight the remarkable ability of wavelet-based dictionaries in achieving highly compressible approximations of compound Poisson processes.

The Critical Smoothness of Generalized Functions

“…This has been generalized for random processes X that are solutions of stochastic differential equations with Lévy white noise W as [28,Corollary 1]. This is consistent with (6) for which γ = 1.…”

“…Perhaps the simplest example of a generalized function for which one can characterize the Besov regularity is the Dirac impulse δ. Its critical function is given by s δ (p) = 1 p − 1 [59, p. 164]; see also [6,Proposition 5] for a wavelet-based proof. From this, one deduce that the critical smoothness of piecewise constant functions f is s f (p) = 1 p .…”

“…The Gaussian white noise W , which is simply the (weak) derivative of the Brownian motion, is strongly related since we have s B (p) = s W (p) + 1 for any p > 0. It has been studied by Veraar over the torus for p ≥ 1 with Fourier domain techniques in [64] and completed for 0 < p < 1 in [6]. These works allow one to deduce that the Brownian motion satisfies…”

“…It therefore comes as no surprise that the critical smoothness function is sufficient to characterize the wavelet compressibility, as detailed in Section 5. The use of the Besov regularity of Lévy white noises and Lévy processes has been used to deduce the rate of the n-term approximation of Lévy processes in [28], with a special emphasis on compound Poisson processes with elementary tools in [6].…”