MULTIVARIATE APPROXIMATION IN TOTAL VARIATION, I: EQUILIBRIUM DISTRIBUTIONS OF MARKOV JUMP PROCESSES

• 成果类型: Article
• 署名作者: Barbour, A. D.; Luczak, M. J.; Xia, A.
• 署名单位: University of Zurich; University of London; Queen Mary University London; University of Melbourne
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1204
• 发表日期: 2018
• 页码: 1351-1404
• 关键词: poisson approximation

摘要

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 Z(d).