Universality of local times of killed and reflected random walks

TL;DR

This paper demonstrates that the local times of killed and reflected random walks exhibit universal asymptotic behaviors, with local times converging to exponential and Bessel process distributions, extending classical results.

Contribution

It extends the second Ray-Knight theorem to all asymptotically stable random walks and establishes the convergence of local times for killed and reflected walks.

Findings

Rescaled local times at point N conditioned on positivity converge to exponential distribution.

Local times of reflected random walks converge in finite-dimensional distributions.

Extension of classical results to a broader class of asymptotically stable random walks.

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(1963) 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.