We are concerned with the equilibrium distribution $\prod _n$ of the $n$th element in a sequence of continuous-time density-dependent Markov processes on the integers. Under a $(2+\alpha )$th moment condition on the jump distributions, we establish a bound of order $O(n^{-(\alpha +1)/2}\sqrt{ \log n})$ on the difference between the point probabilities of $\prod n$ and those of a translated Poisson distribution with the same variance. Except for the factor $\sqrt{ \log n}$, the result is as good as could be obtained in the simpler setting of sums of independent, integer-valued random variables. Our arguments are based on the Stein-Chen method and coupling.

Barbour, A D; Socoll, S (2009). *Local limit approximations for Markov population processes.* Journal of Applied Probability, 46(3):690-708.

## Abstract

We are concerned with the equilibrium distribution $\prod _n$ of the $n$th element in a sequence of continuous-time density-dependent Markov processes on the integers. Under a $(2+\alpha )$th moment condition on the jump distributions, we establish a bound of order $O(n^{-(\alpha +1)/2}\sqrt{ \log n})$ on the difference between the point probabilities of $\prod n$ and those of a translated Poisson distribution with the same variance. Except for the factor $\sqrt{ \log n}$, the result is as good as could be obtained in the simpler setting of sums of independent, integer-valued random variables. Our arguments are based on the Stein-Chen method and coupling.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
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Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | 2009 |

Deposited On: | 31 Dec 2009 10:16 |

Last Modified: | 05 Apr 2016 13:31 |

Publisher: | Applied Probability Trust |

ISSN: | 0021-9002 |

Publisher DOI: | 10.1239/jap/1253279846 |

Related URLs: | http://projecteuclid.org/euclid.jap/1253279846 http://arxiv.org/abs/0902.0886v1 |

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