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Scaling limit of the local time of random walks conditioned to stay positive

Part of: Markov processes Limit theorems Stochastic processes

Published online by Cambridge University Press:  13 February 2024

Wenming Hong  and
Mingyang Sun Open the ORCID record for Mingyang Sun [Opens in a new window]
Wenming Hong*
Affiliation:
Beijing Normal University
Mingyang Sun*
Affiliation:
Beijing Normal University
*
*Postal address: School of Mathematical Sciences and Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing 100875, PR China.
*Postal address: School of Mathematical Sciences and Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing 100875, PR China.
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Abstract

We prove that the local time of random walks conditioned to stay positive converges to the corresponding local time of three-dimensional Bessel processes by proper scaling. Our proof is based on Tanaka’s pathwise construction for conditioned random walks and the derivation of asymptotics for mixed moments of the local time.

Keywords

Random walkBessel processlocal timeconditioning to stay positive

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F17: Functional limit theorems; invariance principles 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 3 , September 2024 , pp. 1060 - 1074
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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