We use cookies to ensure that we give you the best experience on our website. By continuing to browse this repository, you give consent for essential cookies to be used. You can read more about our Privacy and Cookie Policy.

Durham e-Theses
You are in:

On some random walk problems

LO, CHAK,HEI (2017) On some random walk problems. Doctoral thesis, Durham University.

PDF (On some random walk problems) - Accepted Version


We consider several random walk related problems in this thesis. In the first part, we study a Markov chain on $\RP \times S$, where $\RP$ is the non-negative real numbers and $S$ is a finite set, in which when the $\RP$-coordinate is large, the $S$-coordinate of the process is approximately Markov with stationary distribution $\pi_i$ on $S$. Denoting by $\mu_i(x)$ the mean drift of the $\RP$-coordinate of the process at $(x,i) \in \RP \times S$, we give an exhaustive recurrence classification in the case where $\sum_{i} \pi_i \mu_i (x) \to 0$, which is the critical regime for the recurrence-transience phase transition. If $\mu_i(x) \to 0$ for all $i$, it is natural to study the \emph{Lamperti} case where $\mu_i(x) = O(1/x)$; in that case the recurrence classification is known, but we prove new results on existence and non-existence of moments of return times. If $\mu_i (x) \to d_i$ for $d_i \neq 0$ for at least some $i$, then it is natural to study the \emph{generalized Lamperti} case where $\mu_i (x) = d_i + O (1/x)$. 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 $S_n$ on $\R^d$ we study the asymptotic behaviour of the associated centre of mass process $G_n = n^{-1} \sum_{i=1}^n S_i$. 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, $G_n$ is recurrent if $d=1$ and transient if $d \geq 2$. In the transient case we show that $G_n$ 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 $G_n$ is transient in $d=1$.

Item Type:Thesis (Doctoral)
Award:Doctor of Philosophy
Keywords:Random walk; recurrence classification; Non-homogeneous random walk; centre of mass; Lamperti’s problem; Lyapunov function; barycentre; time-average; local central limit theorem; rate of escape.
Faculty and Department:Faculty of Science > Mathematical Sciences, Department of
Thesis Date:2017
Copyright:Copyright of this thesis is held by the author
Deposited On:22 Feb 2018 11:38

Social bookmarking: del.icio.usConnoteaBibSonomyCiteULikeFacebookTwitter