Theory of Probability and Mathematical Statistics
On the local time of a recurrent random walk on $\Z^2$
V. Bohun, A. Marynych
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Abstract: We prove a functional limit theorem for the number of visits by a planar random walk on $\Z^2$ with zero mean and finite second moment to the points of a fixed finite set $P\subset \Z^2$. 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.
Keywords: Extremal process, inverse extremal process, local time, random process with immigration, recurrent planar random walk, simple symmetric planar walk
Bibliography: 1. 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.
2. A. Dvoretzky and P. Erd\H{o}s, Some problems on random walk in space., Proc. Berkeley Sympos. math. Statist. Probability, California, 1951, 353–367.
3. A. Iksanov, Functional limit theorems for renewal shot noise processes with increasing response functions, Stoch. Process. Appl. 123 (2013), no. 6, 1987–2010.
4. – – – – – –, Renewal theory for perturbed random walks and similar processes, Springer International Publishing AG, 2016.
5. 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.
6. 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.
7. – – – – – –, Asymptotics of random processes with immigration I: scaling limits, Bernoulli 23 (2017), 1233–1278.
8. – – – – – –, Asymptotics of random processes with immigration II: convergence to stationarity, Bernoulli 23 (2017), 1279–1298.
9. A. Iksanov and M. M ̈ohle, On the number of jumps of random walks with a barrier, Adv. Appl. Probab. 40 (2008), no. 1, 206–228.
10. 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.
11. Y. Kasahara, 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.
12. Y. Kasahara, 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.
13. A. Marynych and G. Verovkin, 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.
14. E. L. Rva{\v{c}}eva, On domains of attraction of multi-dimensional distributions, Selected Translations in Mathematical Statistics and Probability 2 (1962), 183–205.
15. F. Spitzer, Principles of random walk, vol. 34, Springer Science & Business Media, 2013.
16. 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.
