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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References |
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Additional Information

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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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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*Extremal process as a substitution for “one-sided stable process with index $0$”.*, Stochastic processes and their applications, Proc. Int. Conf., Nagoya, Japan 1985, Lect. Notes Math. 1203, 90–100. MR **872104**
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*A limit theorem for sums of i.i.d. random variables with slowly varying tail probability*, J. Math. Kyoto Univ. **26** (1986), no. 3, 437–443. MR **857228**
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*A functional limit theorem for random processes with immigration in the case of heavy tails*, Modern Stochastics: Theory and Applications **4** (2017), no. 2, 93–108. MR **3668776**
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*On domains of attraction of multi-dimensional distributions*, Selected Translations in Mathematical Statistics and Probability **2** (1962), 183–205. MR **0150795**
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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