Wavelet Compressibility of Compound Poisson Processes
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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 piecewise-constant nature of compound Poisson processes. More precisely, we provide nearly-tight lower and upper bounds for the mean L_2-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.
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