The Annals of Probability

Multivariate approximation in total variation, I: Equilibrium distributions of Markov jump processes

A. D. Barbour, M. J. Luczak, and A. Xia

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For integer valued random variables, the translated Poisson distributions form a flexible family for approximation in total variation, in much the same way that the normal family is used for approximation in Kolmogorov distance. Using the Stein–Chen method, approximation can often be achieved with error bounds of the same order as those for the CLT. In this paper, an analogous theory, again based on Stein’s method, is developed in the multivariate context. The approximating family consists of the equilibrium distributions of a collection of Markov jump processes, whose analogues in one dimension are the immigration-death processes with Poisson distributions as equilibria. The method is illustrated by providing total variation error bounds for the approximation of the equilibrium distribution of one Markov jump process by that of another. In a companion paper, it is shown how to use the method for discrete normal approximation in $\mathbb{Z}^{d}$.

Article information

Ann. Probab., Volume 46, Number 3 (2018), 1351-1404.

Received: December 2015
Revised: December 2016
First available in Project Euclid: 12 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E17: Approximations to distributions (nonasymptotic)
Secondary: 62E20: Asymptotic distribution theory 60J27: Continuous-time Markov processes on discrete state spaces 60C05: Combinatorial probability

Markov jump process multivariate approximation total variation distance infinitesimal generator Stein’s method


Barbour, A. D.; Luczak, M. J.; Xia, A. Multivariate approximation in total variation, I: Equilibrium distributions of Markov jump processes. Ann. Probab. 46 (2018), no. 3, 1351--1404. doi:10.1214/17-AOP1204.

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