Open Access
2016 Universality of local times of killed and reflected random walks
Denis Denisov, Vitali Wachtel
Electron. Commun. Probab. 21: 1-11 (2016). DOI: 10.1214/15-ECP3995


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.


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Denis Denisov. Vitali Wachtel. "Universality of local times of killed and reflected random walks." Electron. Commun. Probab. 21 1 - 11, 2016.


Received: 16 December 2014; Accepted: 21 October 2015; Published: 2016
First available in Project Euclid: 3 February 2016

zbMATH: 1336.60087
MathSciNet: MR3485370
Digital Object Identifier: 10.1214/15-ECP3995

Primary: 60G50
Secondary: 60F17 , 60G40

Keywords: Local time , Random walk , second Ray-Knight theorem

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