## The Annals of Probability

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

#### Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aop/1523520019

Digital Object Identifier
doi:10.1214/17-AOP1204

Mathematical Reviews number (MathSciNet)
MR3785590

Zentralblatt MATH identifier
06894776

#### Citation

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. https://projecteuclid.org/euclid.aop/1523520019

#### References

• Barbour, A. D. (1988). Stein’s method and Poisson process convergence. J. Appl. Probab. Special Vol. 25A 175–184. A celebration of applied probability.
• Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Studies in Probability 2. Oxford Univ. Press, London.
• Barbour, A. D., Luczak, M. J. and Xia, A. (2018). Multivariate approximation in total variation, II: Discrete normal approximation. Ann. Probab. 46 1405–1440.
• Barbour, A. D. and Pollett, P. K. (2012). Total variation approximation for quasi-equilibrium distributions, II. Stochastic Process. Appl. 122 3740–3756.
• Barbour, A. D. and Xia, A. (1999). Poisson perturbations. ESAIM Probab. Stat. 3 131–150.
• Chen, L. H. Y. (1975). Poisson approximation for dependent trials. Ann. Probab. 3 534–545.
• Fang, X. (2014). Discretized normal approximation by Stein’s method. Bernoulli 20 1404–1431.
• Kemeny, J. G. and Snell, J. L. (1960). Finite Markov Chains. Van Nostrand, Princeton, NJ.
• Kemeny, J. G. and Snell, J. L. (1961). Finite continuous time Markov chains. Teor. Verojatnost. i Primenen. 6 110–115.
• Khalil, H. K. (2002). Nonlinear Systems, 3rd ed. Prentice Hall, Upper Saddle River, NJ.
• Presman, E. L. (1983). On the approximation of binomial distributions by means of infinitely divisible ones. Teor. Verojatnost. i Primenen. 28 393–403.
• Roberts, G. O. and Rosenthal, J. S. (1996). Quantitative bounds for convergence rates of continuous time Markov processes. Electron. J. Probab. 1 no. 9, approx. 21 pp.
• Röllin, A. (2005). Approximation of sums of conditionally independent variables by the translated Poisson distribution. Bernoulli 11 1115–1128.
• Röllin, A. (2007). Translated Poisson approximation using exchangeable pair couplings. Ann. Appl. Probab. 17 1596–1614.
• Socoll, S. N. and Barbour, A. D. (2010). Translated Poisson approximation to equilibrium distributions of Markov population processes. Methodol. Comput. Appl. Probab. 12 567–586.
• Tropp, J. A. (2015). Integer factorization of a positive-definite matrix. SIAM J. Discrete Math. 29 1783–1791.