Correlated random walks with a finite memory range (Q2718388)

A one-dimensional correlated random walk \((R_n)_{n\geq 0}\) with (finite) memory range \(M\geq 1\) is given by \(R_n=S_1+S_2+\dots+S_n\), where the increments \(S_1\), \(S_2\), \dots\ are defined as follows: for \(k\leq{M}\), the jumps \(S_k\) are i.i.d.\ symmetric Bernoulli random variables with values~\(\pm 1\); for \(k>M\), a coin showing head with probability \(p\in[0,1]\) is tossed; if there was a head, \(S_k\) takes values \(\pm 1\) with equal probabilities, otherwise, \(S_k=f_{M}(S_{k-1},S_{k-2},\dots,S_{k-M})\), where \(f_{M}\) is a Boolean function from \(\{-1,1\}^M\) onto \(\{-1,1\}\). NEWLINENEWLINENEWLINEA particular case with \(M=1\) was introduced by \textit{G. I. Taylor} in 1921 and was studied by \textit{S. Goldstein} [Q. J. Mech. Appl. Math. 4, 129-156 (1951; Zbl 0045.08102)]. In the present paper, the authors use the transfer-matrix formalism to investigate some cases with \(M>1\) subject to zero mean displacement condition \({\mathbb E}R_n=0\). In particular, they give formulae for the variance \({\mathbb E}(R_n)^2\) and discuss the crossover phenomenon from quadratic to linear behaviour of \({\mathbb E}(R_n)^2\).