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On the sojourn time distribution of a random walk at a multidimensional lattice point. (English. Russian original) Zbl 1495.60077

Theory Probab. Appl. 66, No. 4, 522-536 (2022); translation from Teor. Veroyatn. Primen. 66, No. 4, 657-675 (2021).
Authors’ abstract: “We consider critical symmetric branching random walks on a multidimensional lattice with continuous time and with the source of particle birth and death at the origin. We prove limit theorems on the distribution of the sojourn time of the underlying random walk at a point depending on the lattice dimension under the assumption of finite variance and under a condition leading to infinite variance of jumps. We study the limit distribution of the population of particles at the source for recurrent critical branching random walks.”
The paper is structured in 6 chapters: 1. Introduction (the main results of the paper: Theorem 1 and Theorem 2) – 2. Description of the model (formal description of the model of a BRW) – 3. Auxiliary results (results that are used in the proofs of Theorems 1 and 2) – 4. Proof of Theorem 1 – 5. Proof of Theorem 2 – 6. Distribution of the population size of particles for a recurrent critical BRW – Acknowledgment – References (19 references).

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G50 Sums of independent random variables; random walks
60F17 Functional limit theorems; invariance principles
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References:

[1] F. Spitzer, Principles of Random Walk, The University Series in Higher Mathematics, D. Van Nostrand, Princeton, NJ, 1964. · Zbl 0119.34304
[2] P. Révész, Random Walk in Random and Non-Random Environments, 2nd ed., World Sci. Publ., Hackensack, NJ, 2005, https://doi.org/10.1142/5847. · Zbl 1090.60001
[3] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed., Vol. 2, 2nd ed., John Wiley & Sons, New York, 1968, 1971. · Zbl 0155.23101
[4] K. L. Chung, Markov Chains with Stationary Transition Probabilities, Grundlehren Math. Wiss. 104, Springer-Verlag, Berlin, 1960, https://doi.org/10.1007/978-3-642-49686-8. · Zbl 0092.34304
[5] I. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes, Vols. I, II, Grundlehren Math. Wiss. 210, 218, Springer-Verlag, New York, 1974, 1975. · Zbl 0291.60019
[6] K. L. Chung and G. A. Hunt, On the zeros of \(\sum_1^n\pm 1\), Ann. of Math. (2), 50 (1949), pp. 385-400, https://doi.org/10.2307/1969462. · Zbl 0032.41701
[7] W. Feller, Fluctuation theory of recurrent events, Trans. Amer. Math. Soc., 67 (1949), pp. 98-119, https://doi.org/10.1090/S0002-9947-1949-0032114-7. · Zbl 0039.13301
[8] K. L. Chung, W. H. J. Fuchs, P. Erd\Hos, M. Kac, and M. D. Donsker, Four Papers on Probability, not consecutively paged, Mem. Amer. Math. Soc. 6, Amer. Math. Soc., New York, 1951. · Zbl 1415.60003
[9] K. L. Chung and M. Kac, Remarks on fluctuations of sums of independent random variables, in Four Papers on Probability, Mem. Amer. Math. Soc. 6, Amer. Math. Soc., New York, 1951, https://doi.org/10.1090/memo/0006. · Zbl 0042.37503
[10] R. L. Dobrushin, Two limit theorems for the simplest random walk on a line, Uspekhi Mat. Nauk, 10 (1955), pp. 139-146 (in Russian). · Zbl 0068.32802
[11] G. Kallianpur and H. Robbins, The sequence of sums of independent random variables, Duke Math. J., 21 (1954), pp. 285-307, https://doi.org/10.1215/S0012-7094-54-02128-6. · Zbl 0055.36704
[12] G. Kallianpur and H. Robbins, Ergodic property of the Brownian motion process, Proc. Nat. Acad. Sci. USA, 39 (1953), pp. 525-533, https://doi.org/10.1073/pnas.39.6.525. · Zbl 0053.10003
[13] D. A. Darling, M. Kac, On occupation times for Markoff processes, Trans. Amer. Math. Soc., 84 (1957), pp. 444-458, https://doi.org/10.1090/S0002-9947-1957-0084222-7. · Zbl 0078.32005
[14] E. Yarovaya, Branching random walks with heavy tails, Comm. Statist. Theory Methods, 42 (2013), pp. 3001-3010, https://doi.org/10.1080/03610926.2012.703282. · Zbl 1279.60114
[15] E. B. Yarovaya, Branching Random Walks in an Inhomogeneous Medium, Tsentr Prikladnykh Issledovanii pri Mekhaniko-Matematicheskom Fakul’tete MGU, Moscow, 2007 (in Russian).
[16] E. Vl. Bulinskaya, Hitting times with taboo for a random walk, Siberian Adv. Math., 22 (2012), pp. 227-242, https://doi.org/10.3103/S1055134412040013. · Zbl 1264.60056
[17] A. Rytova and E. Yarovaya, Heavy-tailed branching random walks on multidimensional lattices. A moment approach, Proc. Roy. Soc. Edinburgh Sect. A, 151 (2021), pp. 971-992, https://doi.org/10.1017/prm.2020.46. · Zbl 1478.60238
[18] J. M. Stoyanov, Counterexamples in Probability, 3rd ed., Dover Publications, New York, 2013. · Zbl 1287.60004
[19] H. Pollard, The completely monotonic character of the Mittag-Leffler function, Bull. Amer. Math. Soc., 54 (1948), pp. 1115-1116, https://doi.org/10.1090/S0002-9904-1948-09132-7. · Zbl 0033.35902
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