Subexponential tail asymptotics for a random walk with randomly placed one-way nodes

Abstract

Let p∈]0,1/2] and assign to the integers random variables { ω x }, taking only the two values 1 and p, which serve as an environment. This environment defines a random walk { X n } which, when at x, moves one step to the right with probability ω x , and one step to the left with probability 1− ω x . In particular, at the nodes, i.e., at the locations x with ω x =1, no backtrack is possible. We will assume that the speed v α of the random walk is positive. We then investigate, for v< v α , the decay of the probabilities P ω[X n/n⩽v] (for fixed environment) and P[X n/n⩽v] (averaged over the environment). These probabilities decay subexponentially and there is a wide range of possible normalizations, depending on the distribution of the lengths of the intervals without nodes. We show that in fact only the behaviour of the length of the largest interval without nodes (contained in [0, n]) matters.