Universality of local times of killed and reflected random walks

Denis Denisov,Vitali Wachtel

doi:10.1214/15-ecp3995

Abstract

In this note we first consider local times of random walks killed at leaving positive half-axis. We prove that the distribution of the properly rescaled local time at point $N$ conditioned on being positive converges towards an exponential distribution. The proof is based on known results for conditioned random walks, which allow to determine the asymptotic behaviour of moments of local times. Using this information we also show that the field of local times of a reflected random walk converges in the sense of finite dimensional distributions. This is in the spirit of the seminal result by Knight [10] who has shown that for the symmetric simple random walk local times converge weakly towards a squared Bessel process. Our result can be seen as an extension of the second Ray-Knight theorem to all asymptotically stable random walks.

Highlights

Let {Sn} be a random walk on Z with increments {Xk} which are independent copies of a random variable X
In this note we first consider local times of random walks killed at leaving positive half-axis
We prove that the distribution of the properly rescaled local time at point N conditioned on being positive converges towards an exponential distribution

Summary

Introduction

Let {Sn} be a random walk on Z with increments {Xk} which are independent copies of a random variable X. We shall always assume that X is integrable and EX = 0. This implies that {Sn} is recurrent and, in particular, τ − is almost sure finite. For (α, β) ∈ A and a random variable X write X ∈ D (α, β) if the distribution of X belongs to the domain of attraction of a stable law with characteristic function. This means that there exists an increasing, regularly varying with index 1/α function c(x) such that Sn/c(n) converges in distribution towards the stable law given by (1.1)

Let n

We are interested in the asymptotic behaviour of local times n

This yields

From this we infer that