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

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.