Multivariate approximation in total variation using local dependence

TL;DR

This paper develops new theorems for accurately approximating the distribution of integer-valued random vectors in total variation, especially under local dependence, with applications in graph theory, stochastic processes, and point processes.

Contribution

It introduces two theorems for multivariate normal approximation in total variation, extending previous bounds to unbounded summands with third moment conditions.

Findings

Error bounds comparable to previous work but in total variation

Applicable to local dependence and marked point processes

Demonstrated in four diverse applications

Abstract

We establish two theorems for assessing the accuracy in total variation of multivariate discrete normal approximation to the distribution of an integer valued random vector $W$. The first is for sums of random vectors whose dependence structure is local. The second applies to random vectors~$W$ resulting from integrating the $\mathbb{Z}^d$-valued marks of a marked point process with respect to its ground process. The error bounds are of magnitude comparable to those given in Rinott and Rotar (1996), but now with respect to the stronger total variation distance. Instead of requiring the summands to be bounded, we make third moment assumptions. We demonstrate the use of the theorems in four applications: monochrome edges in vertex coloured graphs, induced triangles and $2$-stars in random geometric graphs, the times spent in different states by an irreducible and aperiodic finite Markov chain, and the maximal points in different regions of a homogeneous Poisson point process.