Random time-changes and asymptotic results for a class of continuous-time Markov chains on integers with alternating rates

TL;DR

This paper analyzes a class of continuous-time Markov chains on integers with alternating transition rates, providing explicit formulas, studying time-changes, and deriving asymptotic results including large deviations.

Contribution

It introduces explicit formulas for probabilities and moments, and explores asymptotic behaviors under various random time-changes, extending previous work in the field.

Findings

Explicit formulas for generating functions, means, variances, and state probabilities.

Analysis of time-changed processes with stable and tempered stable subordinators.

Asymptotic results including large deviations for the process.

Abstract

We consider continuous-time Markov chains on integers which allow transitions to adjacent states only, with alternating rates. We give explicit formulas for probability generating functions, and also for means, variances and state probabilities of the random variables of the process. Moreover we study independent random time-changes with the inverse of the stable subordinator, the stable subordinator and the tempered stable subodinator. We also present some asymptotic results in the fashion of large deviations. These results give some generalizations of those presented in Di Crescenzo A., Macci C., Martinucci B. (2014).