Abstract
An asymptotics is obtained for the distribution tail of the sojourn time for a homogeneous random walk defined on \([0,n]\), above a receding level in a domain of moderate large deviations under Cramér’s condition on the jump distribution.
Similar content being viewed by others
REFERENCES
A. K. AleŢkevičiene, “On the probabilities of large deviations for the maximum of sums of independent random variables,” Teor. Veroaytn. Primen.24, 18 (1979) [Theory Probab. Appl. 24, 16 (1979)].
I. S. Borisov, “A Note on the Distribution of the Number of Crossings of a Strip by a Random Walk,” Teor. Veroyatn. Primen. 53, 345 (2008) [Theory Probab. Appl. 53, 312 (2009)].
I. S. Borisov and A. M. Shoisoronov, “A continuity theorem in the ruin problem,” Sib. Matem. Zh. 52, 765 (2011) [Siberian Math. J. 52, 602 (2011)].
I. S. Borisov and E. I. Shefer, “The asymptotic behavior of the mean sojourn time for a random walk above a receding curvilinear boundary,” Sib. Zh. Chist. i Prikl. Matem.17(4), 18 (2017). [J. Math. Sci. 237, 511 (2019)].
I. S. Borisov and E. I. Shefer, “Asymptotic behavior of the mean sojourn time for a random walk in a domain of large deviations,” Mat. Trudy 22, 3 (2019) [Siberian Adv. Math. 30, 77 (2020)].
A. N. Borodin, “Brownian local time,” Uspekhi Mat. Nauk 44, 7 (1989) [Russian Math. Surveys 44, 1 (1989)].
A. N. Borodin and P. Salminen, Handbook of Brownian Motion – Facts and Formulae (Lan’, St. Petersburg, 2016) [in Russian].
I. I. Gikhman and A. V. Skorokhod, Introduction to Theory of Stochastic Processes (Nauka, Moscow, 1977) [in Russian].
J. Komlos, P. Major, and G. Tusnady, “An approximation of partial sums of independent RV’-s, and the sample DF. II,” Z. Wahrscheinlichkeitstheor. verw. Gebiete34, 33 (1976).
Funding
This work was supported by the Russian Foundation for Basic Research (project No. 18–01–00074 and 19-31-90038) and by the state contract of the Sobolev Institute of Mathematics No. I.1.3 (project No. 0314-2020-0008).
Author information
Authors and Affiliations
Corresponding authors
About this article
Cite this article
Borisov, I.S., Shefer, E.I. Asymptotics of the Distribution Tail of the Sojourn Time for a Random Walk in a Domain of Moderate Large Deviations. Sib. Adv. Math. 30, 162–176 (2020). https://doi.org/10.3103/S1055134420030025
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1055134420030025