Urn-related random walk with drift $\rho x^\alpha / t^\beta$

Mikhail Menshikov,Stanislav Volkov

doi:10.1214/ejp.v13-508

Abstract

We study a one-dimensional random walk whose expected drift depends both on time and the position of a particle. We establish a non-trivial phase transition for the recurrence vs. transience of the walk, and show some interesting applications to Friedman's urn, as well as showing the connection with Lamperti's walk with asymptotically zero drift.

Highlights

Consider the following stochastic processes Xt which may loosely be described as a random walk on R+ with the asymptotic drift given by μt
In our paper we study the random walk whose drift depends both on time and the position of a particle
See Janson [6] and Pemantle [10] for more on urn models

Summary

Introduction

Consider the following stochastic processes Xt which may loosely be described as a random walk on R+ (or in more generality on R) with the asymptotic drift given by μt. The class of stochastic processes we are considering covers simultaneously the Friedman urn, and the walk with an asymptotically zero drift, first probably studied by Lamperti, see [7] and [8]. His one-dimensional walks with drift depending only on the position of the particle naturally arise when proving recurrence of the simple random walk on Z1 and Z2 and transience on Zd, d ≥ 3.

Technical facts

Transience

Recurrence

Special cases

Application to urn models