Multidimensional compound Poisson approximations for symmetric distributions

TL;DR

This work tackles accurate approximation of the distribution of sums of i.i.d. symmetric lattice vectors in multidimensional space by accompanying CP laws and Hipp-type signed measures in total variation. It extends Le Cam’s convolution approach to allow coordinate dependence and derives explicit $d_{TV}$-bounds, alongside Bergström-type long expansions and SCP corrections. The results relax previous requirements (such as axis-aligned support and large zero-probability mass) and include a finite-lines case with sharp rates expressed through sums of variances along each line. Collectively, the paper broadens the applicability of CP approximations in high dimensions, offering practical error controls and higher-order expansion tools for both general and structured (finite-line) distributions.

Abstract

Distribution of the sum of independent identically distributed symmetric lattice vectors is approximated by the accompanying compound Poisson law and the second-order Hipp-type signed compound Poisson measure. Bergström -type asymptotic expansion is constructed. The accuracy of approximation is estimated in the total variation metric and, in many cases, is of the order $O(n^{-1})$.

Theorems & Definitions (26)

• Theorem 3.1
• Example 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Corollary 3.1
• Example 3.2
• Theorem 3.5
• Theorem 3.6
• Corollary 3.2