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The Critical Case of the Cramer-Lundberg Theorem on the Asymptotic Tail Behavior of the Maximum of a Negative Drift Random Walk

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Abstract

We study the asymptotic tail behavior of the maximum M = max{0,S n ,n ≥ = 1} of partial sums S n = ξ1 + ⋯ + ξ n of independent identically distributed random variables ξ12,... with negative mean. We consider the so-called Cramer case when there exists a β > 0 such that E e βξ1 = 1. The celebrated Cramer-Lundberg approximation states the exponential decay of the large deviation probabilities of M provided that Eξ1 e βξ1 is finite. In the present article we basically study the critical case Eξ1 e βξ1 = ∞.

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Additional information

Original Russian Text Copyright © 2005 Korshunov D. A.

The author was supported by the Russian Foundation for Basic Research (Grant 05-01-00810) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-2139.2003.1).

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Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 1335–1340, November–December, 2005.

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Korshunov, D.A. The Critical Case of the Cramer-Lundberg Theorem on the Asymptotic Tail Behavior of the Maximum of a Negative Drift Random Walk. Sib Math J 46, 1077–1081 (2005). https://doi.org/10.1007/s11202-005-0102-2

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Keywords

  • maximum of a random walk
  • probabilities of large deviations
  • light tails
  • exponential change of measure
  • truncated mean value function