Geodesic
On the sojourn time distribution of a random walk at a multidimensional lattice point
Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 4, pp. 657-675 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: branching random walk, multidimensional lattice, particle population distribution at a lattice point, variance of jumps, functional limit theorem, method of moments. 
@article{TVP_2021_66_4_a2,
     author = {A. A. Aparin and G. A. Popov and E. B. Yarovaya},
     title = {On the sojourn time distribution of a~random walk at a~multidimensional lattice point},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {657--675},
     year = {2021},
     volume = {66},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2021_66_4_a2/}
}
TY  - JOUR
AU  - A. A. Aparin
AU  - G. A. Popov
AU  - E. B. Yarovaya
TI  - On the sojourn time distribution of a random walk at a multidimensional lattice point
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2021
SP  - 657
EP  - 675
VL  - 66
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TVP_2021_66_4_a2/
LA  - ru
ID  - TVP_2021_66_4_a2
ER  - 
%0 Journal Article
%A A. A. Aparin
%A G. A. Popov
%A E. B. Yarovaya
%T On the sojourn time distribution of a random walk at a multidimensional lattice point
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2021
%P 657-675
%V 66
%N 4
%U http://geodesic.mathdoc.fr/item/TVP_2021_66_4_a2/
%G ru
%F TVP_2021_66_4_a2
A. A. Aparin; G. A. Popov; E. B. Yarovaya. On the sojourn time distribution of a random walk at a multidimensional lattice point. Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 4, pp. 657-675. http://geodesic.mathdoc.fr/item/TVP_2021_66_4_a2/

[1] F. Spitzer, Principles of random walk, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, NJ–Toronto–London, 1964, xi+406 pp. | MR | Zbl | Zbl

[2] P. Révész, Random walk in random and non-random environments, 2nd ed., World Sci. Publ., Hackensack, NJ, 2005, xvi+380 pp. | DOI | MR | Zbl

[3] V. Feller, Vvedenie v teoriyu veroyatnostei i ee prilozheniya, v. 1, 2, 2-e izd., Mir, M., 1984, 528 s., 752 pp. ; W. Feller, An introduction to probability theory and its applications, т. 1, 3rd ed., John Wiley Sons, Inc., New York–London–Sydney, 1968, xviii+509 с. ; v. 2, 2nd ed., 1971, xxiv+669 pp. | MR | MR | Zbl | MR | Zbl | MR | Zbl

[4] Kai Lai Chung, Markov chains with stationary transition probabilities, Grundlehren Math. Wiss., 104, Springer-Verlag, Berlin–Götingen–Heiderberg, 1960, x+278 pp. | DOI | MR | Zbl | Zbl

[5] I. I. Gihman, A. V. Skorohod, The theory of stochastic processes, v. I, II, Grundlehren Math. Wiss., 210, 218, Springer-Verlag, New York–Heidelberg, 1974, 1975, viii+570 pp., vii+441 pp. | MR | MR | MR | MR | Zbl | Zbl

[6] K. L. Chung, G. A. Hunt, “On the zeros of $\sum_1^n\pm 1$”, Ann. of Math. (2), 50:2 (1949), 385–400 | DOI | MR | Zbl

[7] W. Feller, “Fluctuation theory of recurrent events”, Trans. Amer. Math. Soc., 67 (1949), 98–119 | DOI | MR | Zbl

[8] Kai Lai Chung, W. H. J. Fuchs, P. Erdős, M. Kac, M. D. Donsker, Four papers on probability, Mem. Amer. Math. Soc., 6, Amer. Math. Soc., New York, 1951, 54 pp., not consecutively paged | DOI | Zbl

[9] Kai Lai Chung, M. Kac, “Remarks on fluctuations of sums of independent random variables”, Four papers on probability, Mem. Amer. Math. Soc., 6, Amer. Math. Soc., New York, 1951, 11 pp. | DOI | MR | Zbl

[10] R. L. Dobrushin, “Dve predelnye teoremy dlya prosteishego sluchainogo bluzhdaniya po pryamoi”, UMN, 10:3(65) (1955), 139–146 | MR | Zbl

[11] G. Kallianpur, H. Robbins, “The sequence of sums of independent random variables”, Duke Math. J., 21:2 (1954), 285–307 | DOI | MR | Zbl

[12] G. Kallianpur, H. Robbins, “Ergodic property of the Brownian motion process”, Proc. Nat. Acad. Sci. U.S.A., 39:6 (1953), 525–533 | DOI | MR | Zbl

[13] D. A. Darling, M. Kac, “On occupation times for Markoff processes”, Trans. Amer. Math. Soc., 84:2 (1957), 444–458 | DOI | MR | Zbl

[14] E. Yarovaya, “Branching random walks with heavy tails”, Comm. Statist. Theory Methods, 42:16 (2013), 3001–3010 | DOI | MR | Zbl

[15] E. B. Yarovaya, Vetvyaschiesya sluchainye bluzhdaniya v neodnorodnoi srede, Izd-vo TsPI pri mekh.-matem. f-te MGU, M., 2007, 104 pp.

[16] E. Vl. Bulinskaya, “Hitting times with taboo for a random walk”, Siberian Adv. Math., 22:4 (2012), 227–242 | DOI | MR | Zbl

[17] A. Rytova, E. Yarovaya, “Heavy-tailed branching random walks on multidimensional lattices. A moment approach”, Proc. Roy. Soc. Edinburgh Sect. A, 151:3 (2021), 971–992 | DOI | MR | Zbl

[18] J. M. Stoyanov, Counterexamples in probability, Dover Books on Mathematics, 3rd ed., Dover Publications, Mineola, NY, 2013, xxx+368 pp. | Zbl | Zbl

[19] H. Pollard, “The completely monotonic character of the Mittag-Leffler function”, Bull. Amer. Math. Soc., 54:12 (1948), 1115–1116 | DOI | MR | Zbl