Irregular random walk on nonnegative integers (Q2718056)

The authors consider a Markov chain on \(\mathbb N_0\), called the irregular walk, which makes jumps of length \(m\) to the right with probability \(p\) and jumps of length \(n\) to the left with probability \(1-p\), if the walker's present distance to the origin is not less than \(n\), and to the origin otherwise. The quantities \(m,n\in\mathbb N\) and \(p\in(0,1)\) are parameters in this model. The main result of the paper is the classification of recurrence properties of this walk: it is positive recurrent, null recurrent or transient, according to whether \(p\) is larger than, equal to, or smaller than \(n/(n+m)\). In the case that \(m=2\) and \(n=1\) and that the jump probabilities may depend on the walker's present location, a necessary and sufficient criterion for a given distribution to be the stationary distribution for this walk is given as well. The proofs consist of explicit derivation of the stationary distributions.