Persistent random walks, variable length Markov chains and piecewise deterministic Markov processes
Peggy Cénac, Brigitte Chauvin, Samuel Herrmann, Pierre Vallois
Abstract
A classical random walk
(
S
t
,t∈N)
is defined by
S
t
:=
∑
n=0
t
X
n
, where
(
X
n
)
are i.i.d. When the increments
(
X
n
)
n∈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
)
n≥0
where
(
M
n
)
n≥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.