Multivariate approximation in total variation, I: Equilibrium distributions of Markov jump processes
| Autor: | Aihua Xia, Malwina J. Luczak, Andrew Barbour |
|---|---|
| Přispěvatelé: | University of Zurich |
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Statistics and Probability
62E17 (Primary) 62E20 60J27 60C05 (Secondary) multivariate approximation infinitesimal generator Context (language use) Stein’s method Poisson distribution 01 natural sciences 010104 statistics & probability symbols.namesake Total variation 60J27 510 Mathematics FOS: Mathematics Applied mathematics 60C05 Infinitesimal generator 1804 Statistics Probability and Uncertainty 0101 mathematics 2613 Statistics and Probability Markov jump process Mathematics 62E20 total variation distance 010102 general mathematics Probability (math.PR) Stein's method 10123 Institute of Mathematics Distribution (mathematics) symbols 62E17 Statistics Probability and Uncertainty Random variable Mathematics - Probability Normal family |
| Zdroj: | Ann. Probab. 46, no. 3 (2018), 1351-1404 |
| Popis: | 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 ${\mathbb Z}^d$. Comment: 57 pages. This paper and 1612.07519 together replace an earlier, longer version |
| Databáze: | OpenAIRE |
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