The first returning speed and the last exit speed of a type of Markov chain

TL;DR

This paper investigates the precise rates at which a specific class of Markov chains return to a state and exit from it, depending on the chain's transition probabilities characterized by a sequence {a_n}.

Contribution

It provides detailed analysis of the first return and last exit speeds for a Markov chain with transition probabilities defined by a sequence {a_n}, extending understanding beyond recurrence classification.

Findings

Derived explicit formulas for return and exit speeds based on {a_n}

Identified conditions under which speeds vary significantly

Enhanced understanding of recurrence and transience behaviors

Abstract

Let $\{X_n\}$ be a Markov chain with transition probability $p_{ij}=a_{j-(i-1)^+},\forall i,j\ge 0$, where $a_j=0$ provided $j<0$, $a_0>0$, $a_0+a_1<1$ and $\sum_{n=0}^\infty a_n=1$. Let $\mu=\sum_{n=1}^\infty na_n$. It's known that $\{X_n\}$ is positive recurrent when $\mu<1$; is null recurrent when $\mu=1$; and is transient when $\mu>1$. In this paper, we shall discuss the first returning speed and the last exit speed more precisely by means of $\{a_n\}$