Translated Poisson approximation for Markov chains

TL;DR

This paper develops a translated Poisson approximation method for the distribution of sums of Markov-dependent integer variables, providing error bounds in total variation under certain aperiodicity conditions.

Contribution

It introduces a new approximation technique for Markov-dependent sums and establishes error bounds comparable to normal approximation results.

Findings

Error bounds are comparable to those for normal approximation.

Aperiodicity of the sum between visits is crucial for accurate total variation approximation.

Without aperiodicity, total variation approximation quality deteriorates.

Abstract

The paper is concerned with approximating the distribution of a sum W of n integer valued random variables Y_i, whose distributions depend on the state of an underlying Markov chain X. The approximation is in terms of a translated Poisson distribution, with mean and variance chosen to be close to those of W, and the error is measured with respect to the total variation norm. Error bounds comparable to those found for normal approximation with respect to the weaker Kolmogorov distance are established, provided that the distribution of the sum of the Y_i's between the successive visits of X to a reference state is aperiodic. Without this assumption, approximation in total variation cannot be expected to be good.