Abstract
Let ξ1,ξ2,... be independent random variables with distributions F 1 F 2,... in a triangular array scheme (F i may depend on some parameter). Assume that Eξ i = 0, Eξ 2i < ∞, and put \(S_n = \sum {_{i = 1}^n \;} \xi _i ,\;\overline S _n = \max _{k \leqslant n} S_k\). Assuming further that some regularly varying functions majorize or minorize the “averaged” distribution \(F = \frac{1}{n}\sum {_{i = 1}^n F_i }\), we find upper and lower bounds for the probabilities P(S n > x) and \(P(\bar S_n > x)\). We also study the asymptotics of these probabilities and of the probabilities that a trajectory {S k } crosses the remote boundary {g(k)}; that is, the asymptotics of P(max k≤n (S k − g(k)) > 0). The case n = ∞ is not excluded. We also estimate the distribution of the first crossing time.
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Original Russian Text Copyright © 2005 Borovkov A. A.
The author was supported by the Russian Foundation for Basic Research (Grant 02-01-00902) and the Grant Council of the President of the Russian Federation (Grant NSh-2139.2003.1).
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Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 1265–1287, November–December, 2005.
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Borovkov, A.A. Asymptotic Analysis for Random Walks with Nonidentically Distributed Jumps Having Finite Variance. Sib Math J 46, 1020–1038 (2005). https://doi.org/10.1007/s11202-005-0097-8
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DOI: https://doi.org/10.1007/s11202-005-0097-8