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.

中文翻译：

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复合泊松过程的小波压缩性

在本文中，我们描述了复合泊松过程的小波压缩性。为此，我们在 Haar 小波基上扩展给定的复合泊松过程并分析其渐近逼近性质。通过仅考虑达到给定尺度的非零小波系数，我们称之为稀疏近似，我们利用了源自复合泊松过程的分段常数性质的小波展开的极端稀疏性。更准确地说，我们为复合泊松过程的平均 $L_2$-稀疏近似误差提供了近乎严格的下限和上限。使用这些边界，我们然后证明稀疏近似误差具有次指数和超多项式渐近行为。我们通过复合泊松过程的数值模拟来说明这些理论结果。