Bounds for mixing times for finite semi-Markov processes with heavy-tail jump distribution

TL;DR

This paper investigates how the mixing times of finite Markov chains are affected when time is replaced by a renewal process with heavy-tailed inter-arrival times, especially focusing on Mittag-Leffler distributions with infinite mean.

Contribution

It provides bounds on mixing times for semi-Markov processes with heavy-tail inter-arrival times, particularly for Mittag-Leffler distributions with infinite expectation.

Findings

Derived bounds for mixing times under heavy-tailed renewal processes

Analyzed the impact of Mittag-Leffler distributed inter-arrival times on mixing behavior

Extended understanding of semi-Markov processes with infinite mean waiting times

Abstract

Consider a Markov chain with finite state space and suppose you wish to change time replacing the integer step index $n$ with a random counting process $N(t)$. What happens to the mixing time of the Markov chain? We present a partial reply in a particular case of interest in which $N(t)$ is a counting renewal process with power-law distributed inter-arrival times of index $\beta$. We then focus on $\beta \in (0,1)$, leading to infinite expectation for inter-arrival times and further study the situation in which inter-arrival times follow the Mittag-Leffler distribution of order $\beta$.