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May 2018 Multivariate approximation in total variation, I: Equilibrium distributions of Markov jump processes
A. D. Barbour, M. J. Luczak, A. Xia
Ann. Probab. 46(3): 1351-1404 (May 2018). DOI: 10.1214/17-AOP1204


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 $\mathbb{Z}^{d}$.


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A. D. Barbour. M. J. Luczak. A. Xia. "Multivariate approximation in total variation, I: Equilibrium distributions of Markov jump processes." Ann. Probab. 46 (3) 1351 - 1404, May 2018.


Received: 1 December 2015; Revised: 1 December 2016; Published: May 2018
First available in Project Euclid: 12 April 2018

zbMATH: 06894776
MathSciNet: MR3785590
Digital Object Identifier: 10.1214/17-AOP1204

Primary: 62E17
Secondary: 60C05 , 60J27 , 62E20

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

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 3 • May 2018
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