Refined total variation bounds in the multivariate and compound Poisson approximation

Abstract

We consider the approximation of a convolution of possibly different probability measures by (compound) Poisson distributions and also by related signed measures of higher order. We present new total variation bounds having a better structure than those from the literature. A numerical example illustrates the usefulness of the bounds, and an application in the Poisson process approximation is given. The proofs use arguments from Kerstan (1964) and Roos (1999b) in combination with new smoothness inequalities, which could be of independent interest.2000 Mathematics Subject Classification. Primary 60F05; secondary 60G50, 62E17.

Compound Poisson approximations in $\ell_p$-norm for sums of weakly dependent vectors

“…The upper bound follows directly from one-dimensional Poisson approximation to the binomial distribution; see [4], p. 29 and Equation (1.1) in [16] . The lower bound is a special case of Proposition 1.3 in [16] .…”

“…The upper bound follows directly from one-dimensional Poisson approximation to the binomial distribution; see [4], p. 29 and Equation (1.1) in [16] . The lower bound is a special case of Proposition 1.3 in [16] . Note that it is impossible to switch from k-dimensional case to 1-dimensional, if some dependence of rvs exits.…”

“…For compound Poisson approximations in Lévy, Lévy -Prokhorov and Kolmogorov metrics, see [19,20,21]. Poisson approximation in total variation for sums of independent lattice vectors concentrated on standard basis of R k was considered in [1,11,13,15,16]. In [11,13] local point metric was also used.…”

Multivariate Poisson and Poisson process approximations with applications to Bernoulli sums and $U$-statistics

“…, x d ) ∈ R d . The accuracy of the multivariate Poisson approximation has mostly been studied in terms of the total variation distance; among others we mention [1,3,4,6,13,26,27]. In contrast, we consider the Wasserstein distance.…”

“…The multivariate Poisson approximation of the multinomial distribution, and more generally of the sum of independent Bernoulli random vectors, has already been tackled by many authors in terms of the total variation distance. Among others, we refer the reader to [4,12,25,27] and the survey [21]. Unlike the mentioned papers, we assume that Y (1) , .…”