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.
References
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References
- N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. MR 898871
- A. Dvoretzky and P. Erdős, Some problems on random walk in space., Proc. Berkeley Sympos. math. Statist. Probability, California, 1951, 353–367. MR 0047272
- A. Iksanov, Functional limit theorems for renewal shot noise processes with increasing response functions, Stoch. Process. Appl. 123 (2013), no. 6, 1987–2010. MR 3038496
- —, Renewal theory for perturbed random walks and similar processes, Springer International Publishing AG, 2016. MR 3585464
- A. Iksanov, Z. Kabluchko, A. Marynych, and G. Shevchenko, Fractionally integrated inverse stable subordinators, Stochastic Processes and their Applications 127 (2017), no. 1, 80–106. MR 3575536
- A. Iksanov, A. Marynych, and M. Meiners, Limit theorems for renewal shot noise processes with eventually decreasing response functions, Stoch. Process. Appl. 124 (2014), no. 6, 2132–2170. MR 3188351
- —, Asymptotics of random processes with immigration I: scaling limits, Bernoulli 23 (2017), 1233–1278. MR 3606765
- —, Asymptotics of random processes with immigration II: convergence to stationarity, Bernoulli 23 (2017), 1279–1298. MR 3606766
- A. Iksanov and M. Möhle, On the number of jumps of random walks with a barrier, Adv. Appl. Probab. 40 (2008), no. 1, 206–228. MR 2411821
- Z. Kabluchko and A. Marynych, Renewal shot noise processes in the case of slowly varying tails, Theory of Stochastic Processes 21 (2016), no. 2, 14–21. MR 3662592
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- W. Whitt, Stochastic-process limits. an introduction to stochastic-process limits and their application to queues, Springer Series in Operations Research, Springer-Verlag, New York, 2002. MR 1876437
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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
