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Article overview
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Persistent random walks, variable length Markov chains and piecewise deterministic Markov processes | | Peggy Cénac
; Brigitte Chauvin
; Samuel Herrmann
; Pierre Vallois
; | | Date: |
16 Aug 2012 | | Abstract: | A classical random walk $(S_t, tinmathbb{N})$ is defined by
$S_t:=displaystylesum_{n=0}^t X_n$, where $(X_n)$ are i.i.d. When the
increments $(X_n)_{ninmathbb{N}}$ 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 Herrmann and Vallois,
2010; 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 $(X_n,M_n)_{nge 0}$ where $(M_n)_{nge 0}$ 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. | | Source: | arXiv, 1208.3358 | | Services: | Forum | Review | PDF | Favorites | |
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