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Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

   
 
 

 

On the local time of a recurrent random walk on $\mathbb {Z}^2$


Authors: V. Bohun and A. Marynych
Journal: Theor. Probability and Math. Statist. 105 (2021), 69-78
MSC (2020): Primary 60F05; Secondary 60K05
DOI: https://doi.org/10.1090/tpms/1156
Published electronically: December 7, 2021
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Abstract: We prove a functional limit theorem for the number of visits by a planar random walk on $\mathbb {Z}^2$ with zero mean and finite second moment to the points of a fixed finite set $P\subset \mathbb {Z}^2$. The proof is based on the analysis of an accompanying random process with immigration at renewal epochs in case when the inter-arrival distribution has a slowly varying tail.


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Additional Information

V. Bohun
Affiliation: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine
Email: vladyslavbogun@gmail.com

A. Marynych
Affiliation: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine
MR Author ID: 848771
Email: marynych@unicyb.kiev.ua

Keywords: Extremal process, inverse extremal process, local time, random process with immigration, recurrent planar random walk, simple symmetric planar walk
Received by editor(s): June 29, 2021
Published electronically: December 7, 2021
Additional Notes: This work was supported by National Research Foundation of Ukraine (project 2020.02/0014 “Asymptotic regimes of perturbed random walks: on the edge of modern and classical probability”)
Article copyright: © Copyright 2021 Taras Shevchenko National University of Kyiv