On the Convergence Rate in a Local Renewal Theorem for a Random Markov Walk

IF 0.6 4区数学 Q3 MATHEMATICS

Mathematical Notes Pub Date : 2024-07-05 DOI:10.1134/s0001434624030209

G. A. Bakai

引用次数: 0

Abstract

Suppose that a sequence $\{X_n\}_{n\ge 0}$ of random variables is a homogeneous irreducible Markov chain with finite set of states. Let $\xi_n$, $n\in\mathbb{N}$, be random variables defined on the chain transitions.

The renewal function

$$u_k:=\sum_{n=0}^{+\infty} \mathsf P(S_n=k), \qquad k\in\mathbb{N},$$

where $S_0:=0$ and $S_n:=\xi_1+\dots + \xi_n$, $n\in\mathbb{N}$, is introduced. It is shown that this function converges to its limit at an exponential rate, and an explicit description of the exponent is given.

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论随机马尔可夫散步局部更新定理的收敛率

Abstract 假设随机变量序列 $\{X_n\}_{n\ge 0}$是一个具有有限状态集的同构不可还原马尔可夫链。让 $\xi_n$, $n\in\mathbb{N}$, 都是定义在链转换上的随机变量。更新函数 $$u_k:=\sum_{n=0}^{+\infty}\其中，引入了$S_0:=0$和$S_n:=\xi_1+\dots +\xi_n$, \(nin\mathbb{N})。结果表明，该函数以指数速度收敛到其极限，并给出了指数的明确描述。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Notes 数学-数学

CiteScore

0.90

自引率

16.70%

发文量

179

审稿时长

24 months

期刊介绍： Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.