Persistent Random Walks. II. Functional Scaling Limits

Abstract

We give a complete and unified description -under some stability assumptions -of the functional scaling limits associated with some persistent random walks for which the recurrent or transient type is studied in [1]. As a result, we highlight a phase transition phenomenon with respect to the memory. It turns out that the limit process is either Markovian or not according to -to put it in a nutshell -the rate of decrease of the distribution tails corresponding to the persistent times. In the memoryless situation, the limits are classical strictly stable Lévy processes of infinite variations. However, we point out that the description of the critical Cauchy case fills some lacuna even in the closely related context of Directionally Reinforced Random Walks (DRRWs) for which it has not been considered yet. Besides, we need to introduced some relevant generalized drift -extended the classical one -in order to study the critical case but also the situation when the limit is no longer Markovian. It appears to be in full generality a drift in mean for the Persistent Random Walk (PRW). The limit processes keeping some memory -given by some variable length Markov chain -of the underlying PRW are called arcsine Lamperti anomalous diffusions due to their marginal distribution which are computed explicitly here. To this end, we make the connection with the governing equations for Lévy walks, the occupation times of skew Bessel processes and a more general class modelled on Lamperti processes. We also stress that we clarify some misunderstanding regarding this marginal distribution in the framework of DRRWs. Finally, we stress that the latter situation is more flexible -as in the first paper -in the sense that the results can be easily generalized to a wider class of PRWs without renewal pattern.

Local limit theorem for a Markov additive process on with a null recurrent internal Markov chain

“…First, we can consider spatial inhomogeneity by weakening the group structure, replacing it, for instance, by a directed graph as in [7,9,10,12,24,33,34,54]. Secondly, we can study temporal inhomogeneous random walks by introducing a notion of memory as in the model of reinforced [53,64], excited [5,58], self-interacting [23,55], or persistent random walks [17,18,19,20,21], or also the Markov additive processes that are at the core of this paper. All these models belong to the larger class of stochastic processes with long range dependency.…”

Non-regular g-measures and variable length memory chains

“…This class has been also used as a tool to study general g-measures, for instance in Gallo and Garcia (2013), Gallo and Paccaut (2013), Garivier (2015), Oliveira (2015). Finally, some recent works have explored the relation with dynamical systems (Cénac et al 2012) or used this class to create interesting random walk models (Cénac et al 2018, Cénac et al 2020, Cénac et al 2019.…”

Non-regular g-measures and variable length memory chains

“…This class has been also used as a tool to study general g-measures, for instance in Gallo & Garcia (2013); Gallo & Paccaut (2013); Garivier (2015); Oliveira (2015). Finally, some recent works have explored the relation with dynamical systems (Cénac et al, 2012; or used this class to create interesting random walk models (Cénac et al, , 2019.…”