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

3 BIGS - Biology, genetics and statistics
Inria Nancy - Grand Est, IECL - Institut Élie Cartan de Lorraine
Abstract : A classical random walk $(S_t,\, t\in\mathbb{N})$ is defined by $S_t:=\displaystyle\sum_{n=0}^t X_n$, where $(X_n)$ are i.i.d. When the increments $(X_n)_{n\in\mathbb{N}}$ are a one-order Markov chain, a short memory is introduced in the dynamics of $(S_t)$. This so-called ''persistent'' random walk is 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 \cite{Herrmann-Vallois, Tapiero-Vallois, Tapiero-Vallois2}). 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.
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https://hal.archives-ouvertes.fr/hal-00719450
Contributor : Peggy Cenac <>
Submitted on : Friday, July 20, 2012 - 10:43:35 AM
Last modification on : Thursday, February 7, 2019 - 4:33:17 PM
Long-term archiving on : Friday, March 31, 2017 - 10:58:16 AM

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• HAL Id : hal-00719450, version 2

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Peggy Cenac, Brigitte Chauvin, Samuel Herrmann, Pierre Vallois. Persistent random walks, variable length Markov chains and piecewise deterministic Markov processes. 2012. ⟨hal-00719450v2⟩

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