The paper is concerned with approximating the distribution of a sum W of integer valued random variables Y i , 1 ≤ i ≤ n, whose distributions depend on the state of an underlying Markov chain X. The approximation is in terms of a translated Poisson distribution, with mean and variance chosen to be close to those of W, and the error is measured with respect to the total variation norm. Error bounds comparable to those found for normal approximation with respect to the weaker Kolmogorov distance are established, provided that the distribution of the sum of the Y i ’s between the successive visits of X to a reference state is aperiodic. Without this assumption, approximation in total variation cannot be expected to be good.

Barbour, A D; Lindvall, T (2009). *Erratum: Translated Poisson approximation for Markov chains.* Journal of Theoretical Probability, 22(1):279-280.

## Abstract

The paper is concerned with approximating the distribution of a sum W of integer valued random variables Y i , 1 ≤ i ≤ n, whose distributions depend on the state of an underlying Markov chain X. The approximation is in terms of a translated Poisson distribution, with mean and variance chosen to be close to those of W, and the error is measured with respect to the total variation norm. Error bounds comparable to those found for normal approximation with respect to the weaker Kolmogorov distance are established, provided that the distribution of the sum of the Y i ’s between the successive visits of X to a reference state is aperiodic. Without this assumption, approximation in total variation cannot be expected to be good.

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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: | 19 Feb 2010 10:15 |

Last Modified: | 05 Apr 2016 13:22 |

Publisher: | Springer |

ISSN: | 0894-9840 |

Publisher DOI: | https://doi.org/10.1007/s10959-008-0190-6 |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=2472017 |

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