## Аннотация

We solve the problem of asymptotic behaviour of the renewal measure (Green function) generated by a transient Lamperti's Markov chain X_{n} in R, that is, when the drift of the chain tends to zero at infinity. Under this setting, the average time spent by X_{n} in the interval (x, x + 1] is roughly speaking the reciprocal of the drift and tends to infinity as x grows. For the first time we present a general approach relying on a diffusion approximation to prove renewal theorems for Markov chains. We apply a martingale-type technique and show that the asymptotic behaviour of the renewal measure heavily depends on the rate at which the drift vanishes. The two main cases are distinguished, either the drift of the chain decreases as 1/x or much slower than that, say as 1/x^{α} for some α ∈ (0, 1). The intuition behind how the renewal measure behaves in these two cases is totally different. While in the first case X_{n}^{2} /n converges weakly to a Γ-distribution and there is no law of large numbers available, in the second case a strong law of large numbers holds true for X_{n}^{1+α}/n and further normal approximation is available.

Язык оригинала | английский |
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Страницы (с-по) | 7253-7286 |

Число страниц | 34 |

Журнал | Transactions of the American Mathematical Society |

Том | 373 |

Номер выпуска | 10 |

DOI | |

Состояние | Опубликовано - окт 2020 |

Опубликовано для внешнего пользования | Да |