Persistent random walks , variable length Markov chains and piecewise deterministic Markov processes ∗

A classical random walk (St, t ∈ N) is defined by St := t ∑ n=0 Xn, where (Xn) are i.i.d. When the increments (Xn)n∈N are a one-order Markov chain, a short memory is introduced in the dynamics of (St). This so-called “persistent” random walk is nolonger Markovian and, under suitable conditions, the rescaled process converges towards the integrated telegraph noise (ITN) as the time-scale and space-scale parameters tend to zero (see [11, 17, 18]). The ITN process is effectively non-Markovian too. The aim is to consider persistent random walks (St) whose increments are Markov chains with variable order which can be infinite. This variable memory is enlighted by a one-to-one correspondence between (Xn) and a suitable Variable Length Markov Chain (VLMC), since for a VLMC the dependency from the past can be unbounded. The key fact is to consider the non Markovian letter process (Xn) as the margin of a couple (Xn,Mn)n≥0 where (Mn)n≥0 stands for the memory of the process (Xn). We prove that, under a suitable rescaling, (Sn, Xn,Mn) converges in distribution towards a time continuous process (S(t), X(t),M(t)). The process (S(t)) is a semi-Markov and Piecewise Deterministic Markov Process whose paths are piecewise linear. 2010 Mathematics Subject Classification. 60J10, 60J27, 60F05, 60G17, 60G40, 60K15.