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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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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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  1. 1.

    Cramer H., Collective Risk Theory, Esselte, Stockholm (1955).

  2. 2.

    Asmussen S., Applied Probability and Queues, Springer-Verlag, New York (2003).

  3. 3.

    Borovkov A. A., Stochastic Processes in Queueing Theory, Springer-Verlag, New York; Berlin (1976).

  4. 4.

    Feller W., An Introduction to Probability Theory and Its Applications. Vol. 2, John Wiley, New York (1971).

  5. 5.

    Korshunov D., “On the distribution tail of the maximum of a random walk,” Stochastic Process. Appl., 72, 97–103 (1997).

  6. 6.

    Bingham N. H., Goldie C. M., and Teugels J. L., Regular Variation, Cambridge Univ. Press, Cambridge (1987).

  7. 7.

    Denisov D., Foss S., and Korshunov D., “Tail asymptotics for the supremum of a random walk when the mean is not finite,” Queueing Systems, 46, 15–33 (2004).

  8. 8.

    Erickson K. B., “Strong renewal theorems with infinite mean,” Trans. Amer. Math. Soc., 151, 263–291 (1970).

  9. 9.

    Erickson K. B., “A renewal theorem for distributions on R 1 without expectation,” Bull. Amer. Math. Soc., 77, 406–410 (1971).

  10. 10.

    Garsia A. and Lamperti J., “A discrete renewal theorem with infinite mean,” Comment. Math. Helv., 37, 221–234 (1963).

  11. 11.

    Williamson J. A., “Random walks and Riesz kernels,” Pacific J. Math., 25, 393–415 (1968).

  12. 12.

    deBruijn N. G. and Erdos P., “On a recursion formula and some Tauberian theorems,” J. Res. Nat. Bur. Standards, 50, 161–164 (1953).

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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).


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