2012/08/16 by Peggy Cénac, Brigitte Chauvin, Cénac, Peggy +5 · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR
paper · pdf · doi:10.48550/arxiv.1208.3358
arxiv created 2012/08/16 · openalex publication_date 2012/08/16 · arxiv updated 2012/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A classical random walk (St, t∈ℕ) is defined by St:=∑n=0t 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 Herrmann and Vallois, 2010; Tapiero-Vallois, Tapiero-Vallois2). 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 (S0(t),X(t),M(t)). The process (S0(t)) is a semi-Markov and Piecewise Deterministic Markov Process whose paths are piecewise linear.