On some random walk problems

Abstract

We consider several random walk related problems in this thesis. In the first part, we study a Markov chain on , where is the non-negative real numbers and is a finite set, in which when the -coordinate is large, the -coordinate of the process is approximately Markov with stationary distribution on . Denoting by the mean drift of the -coordinate of the process at , we give an exhaustive recurrence classification in the case where , which is the critical regime for the recurrence-transience phase transition. If for all , it is natural to study the case where ; in that case the recurrence classification is known, but we prove new results on existence and non-existence of moments of return times. If for for at least some , then it is natural to study the case where . By exploiting a transformation which maps the generalized Lamperti case to the Lamperti case, we obtain a recurrence classification and an existence of moments result for the former. The generalized Lamperti case is seen to be more subtle, as the recurrence classification depends on correlation terms between the two coordinates of the process.

In the second part of the thesis, for a random walk on we study the asymptotic behaviour of the associated centre of mass process . For lattice distributions we give conditions for a local limit theorem to hold. We prove that if the increments of the walk have zero mean and finite second moment, is recurrent if and transient if . In the transient case we show that has diffusive rate of escape. These results extend work of Grill, who considered simple symmetric random walk. We also give a class of random walks with symmetric heavy-tailed increments for which is transient in .