Non-Markovian chains with long-range dependence and their scaling limits

TL;DR

This paper explores non-Markovian chains with long-range dependence, extending existing models by introducing dependence among waiting times and unifying them through a general time-change framework.

Contribution

It introduces new non-Markovian chain models with dependent waiting times and unifies them via a comprehensive time-change theory for Markov chains.

Findings

Para-Markov chains with Schur-constant dependence are characterized.

Time-changed Markov chains with inverse stable processes are analyzed.

The models generalize classical semi-Markov processes.

Abstract

There is a well-established theory linking certain semi-Markov chains and continuous-time random walks to time-fractional equations and anomalous diffusion. In this work, we go beyond the semi-Markov framework by considering some non-Markovian chains, which exhibit long-memory behaviour, due to stochastic dependence among their waiting times. Particular attention is devoted to the so-called para-Markov chains. Their waiting times share the same marginal distributions as those of the above mentioned semi-Markov chains, but they are dependent; their joint distribution is of Schur-constant type and is closely related to complete Bernstein functions and De Finetti's theorems. A second model that we focus on is given by time-changed Markov chains, where the random time is the inverse of an increasing stable process. This generalizes well-known semi-Markov models available in the literature, which typically focus solely on the inverse of the Levy stable subordinator. The above mentioned models are unified by a general theory of time change of Markov chains.