Abstract
Suppose that ξ, ξ(1), ξ(2), ... are independent identically distributed random variables such that −ξ is semiexponential; i.e., \(P( - \xi \geqslant t) = e^{ - t^\beta L(t)} \) is a slowly varying function as t → ∞ possessing some smoothness properties. Let E ξ = 0, D ξ = 1, and S(k) = ξ(1) + ⋯ + ξ(k). Given d > 0, define the first upcrossing time η +(u) = inf{k ≥ 1: S(k) + kd > u} at nonnegative level u ≥ 0 of the walk S(k) + kd with positive drift d > 0. We prove that, under general conditions, the following relation is valid for \(u = (n) \in \left[ {0, dn - N_n \sqrt n } \right]\):
, where x = u − nd < 0 and an arbitrary fixed sequence N n not exceeding \(d\sqrt n \) tends to ∞.
The conditions under which we prove (0.1) coincide exactly with the conditions under which the asymptotic behavior of the probability P(S(n) ≤ x) for \(x \leqslant - \sqrt n \) was found in [1] (for \(x \in \left[ { - \sqrt n ,0} \right]\) it follows from the central limit theorem).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Borovkov A. A. and Mogul’skii A. A., “Integro-local and integral theorems for sums of random variables with semiexponential distributions,” Siberian Math. J., 47, No. 6, 990–1026 (2006).
Doney R. A., “On the asymptotic behavior of the first passage times for transient random walk,” Probab. Theory Related Fields, 81, No. 2, 239–246 (1989).
Borovkov A. A., “On the asymptotics of distributions of the first passage times. I,” Mat. Zametki, 75, No. 1, 24–39 (2004).
Borovkov A. A., “On the asymptotics of distributions of the first passage times. II,” Mat. Zametki, 75, No. 3, 350–359 (2004).
Borovkov A. A., “Limit theorems on the distribution of maxima of sums of bounded lattice random variables. I,” Theory Probab. Appl., 5, 125–155 (1960).
Borovkov A. A., “New limit theorems for boundary problems for sums of independent summands,” Sibirsk. Mat. Zh., 3, No. 5, 654–694 (1962).
Mogul’skii A. A. and Rogozin B. A., “Random walks in a positive quadrant. III: Constants in integral and local theorems,” Mat. Trudy, 4, No. 1, 68–93 (2001).
Mogul’skii A. A. and Rogozin B. A., “A local theorem for the first hitting time of a fixed level by a random walk,” Siberian Adv. Math., 15, No. 3, 1–27 (2005).
Borovkov A. A. and Borovkov K. A., Asymptotic Analysis of Random Walks. Part I: Slowly Decreasing Jumps [in Russian], Nauka, Moscow (to appear).
Bertoin J. and Doney R. A., “Some asymptotic results for transient random walks,” Adv. in Appl. Probab., 28, No. 1, 207–226 (1996).
Pinelis I. F., “A problem on large deviations in a space of trajectories,” Theory Probab. Appl., 26, 69–84 (1981).
Rozovskii L. V., “Probabilities of large deviations of sums of independent random variables with a common distribution function that belongs to the domain of attraction of a normal law,” Theory Probab. Appl., 34, No. 4, 625–644 (1989).
Asmussen S., Kluppelberg C., and Sigman K., “Sampling at sub-exponential times, with queueing applications,” Stochastic Process. Appl., 79, 265–286 (1999).
Baltrunas A., Daley D., and Kluppelberg C., “Tail behaviour of the busy period of a GI/GI/1S queue with subexponential service times,” Stochastic Process. Appl., 111, 237–258 (2004).
Jelenkovich P. R. and Momucilovic P., “Large deviations of square root insensitive random sums,” Math. Oper. Res., 29, No. 2, 398–406 (2004).
Denisov D., “Asymptotics for first passage times of Levi processes and random walks,” in: Abstracts: IV Intern. Conf. “Limit Theorems in Probability Theory and Their Applications,” Novosibirsk, 2006, p. 14.
Denisov D. and Shneer V., “Asymptotics for first passage times of Levi processes and random walks,” Ann. Appl. Prob. (to appear).
Borovkov A. A., “Large deviation probabilities for random walks with semiexponential distributions,” Siberian Math. J., 41, No. 6, 1061–1093 (2000).
Rozovskii L. V., “Probabilities of large deviations on the whole axis,” Teor. Veroyatnost. i Primenen., 38, No. 1, 79–109 (1993).
Mikosh T. and Nagaev A. V., “Large deviations of heavy-tailed sums with applications in insurance,” Extremes, 1, No. 1, 81–110 (1998).
Nagaev A. V., “Integral limit theorems taking deviations into account when Cramér’s condition does not hold,” Teor. Veroyatnost. i Primenen., 14, I: No. 1, 51–64; II: No 2, 203–214 (1969).
Ibragimov I. A. and Linnik Yu. V., Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing Company, Groningen (1971).
Feller W., An Introduction to Probability Theory and Its Applications. Vol. 2 [Russian translation], Mir, Moscow (1984).
Borovkov A. A., “On the limit conditional distributions connected with large deviations,” Siberian Math. J., 37, No. 4, 635–646 (1996).
Borovkov A. A. and Mogul’skii A. A., “Integro-local theorems for sums of random vectors in a series scheme,” Mat. Zametki, 79, No. 4, 505–521 (2006).
Borovkov A. A., Stochastic Processes in Queueing Theory, Springer, New York etc. (1976).
Author information
Authors and Affiliations
Additional information
Original Russian Text Copyright © 2006 Mogul’skiĭ A. A.
__________
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 47, No. 6, pp. 1323–1341, November–December, 2006.
Rights and permissions
About this article
Cite this article
Mogul’skii, A.A. Large deviations of the first passage time for a random walk with semiexponentially distributed jumps. Sib Math J 47, 1084–1101 (2006). https://doi.org/10.1007/s11202-006-0117-3
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11202-006-0117-3