Multidimensional compound Poisson approximations for symmetric distributions

TL;DR

This paper develops advanced approximation methods for symmetric lattice vector sums using compound Poisson laws, providing precise asymptotic expansions and accuracy estimates in total variation.

Contribution

It introduces a second-order asymptotic expansion for symmetric distributions, improving approximation accuracy with explicit error bounds.

Findings

Approximation accuracy often of order O(n^{-1}) in total variation

Constructs Bergström-type asymptotic expansions for symmetric sums

Provides explicit error estimates for compound Poisson approximations

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. Bergstr\"om -type asymptotic expansion is constructed. The accuracy of approximation is estimated in the total variation metric and, in many cases, is of the order $O(n^{-1})$.