Persistent Random Walks, Variable Length Markov Chains and Piecewise Deterministic Markov Processes

#### P. Cenac, B. Chauvin, S. Herrmann, P. Vallois

2013, v.19, Issue 1, 1-50

ABSTRACT

A classical random walk $(S_t, t\in N)$ is defined by $S_t := \sum_{n=0}^t X_n$, where $(X_n)$ are i.i.d. When the increments $(X_n)_{n\in N}$ are a one-order Markov chain, a short memory is introduced in the dynamics of $(S_t)$. This so-called "persistent" random walk is no longer 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 S. Herrmann and P. Vallois, From persistent random walk to the telegraph noise. Stoch. and Dynamics, 2010, v. 10, N 2, 161-196; C.S. Tapiero and P. Vallois, Memory-based persistence in a counting random walk process. Physica A, 2007, v. 386, N 1, 303-307; C.S. Tapiero and P. Vallois, A claims persistence process and insurance. J. Insurance Math. Econom., 2009, v. 44, N 3, 367-373). The ITN process is effectively non-Markovian too. The aim is to consider persistent random walks $(S_t)$ whose increments are Markov chains with variable order which can be infinite. This variable memory is enlighted by a one-to-one correspondence between $(X_n)$ 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 $(X_n)$ as the margin of a couple $(X_n,M_n)_{n\ge 0}$ where $(M_n)_{n\ge 0}$ stands for the memory of the process $(X_n)$. We prove that, under a suitable rescaling, $(S_n,X_n,M_n)$ converges in distribution towards a time continuous process $(S^0(t),X(t),M(t))$. The process $(S^0(t))$ is a semi-Markov and Piecewise Deterministic Markov Process whose paths are piecewise linear.

Keywords: persistent random walk,variable length Markov chain,integrated telegraph noise,piecewise deterministic Markov processes,semi Markov processes,variable memory,simple and double infinite combs