  In this paper, we precisely quantify the wavelet compressibility of compound
Poisson processes. To that end, we expand the given random process over the
Haar wavelet basis and we analyse its asymptotic approximation properties. By
only considering the nonzero wavelet coefficients up to a given scale, what we
call the greedy 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 lower and upper bounds for the mean
squared error of greedy approximation of compound Poisson processes. We are
then able to deduce that the greedy approximation error has a sub-exponential
and super-polynomial asymptotic behavior. Finally, we provide numerical
experiments to highlight the remarkable ability of wavelet-based dictionaries
in achieving highly compressible approximations of compound Poisson processes.
