Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T14:01:21.180Z Has data issue: false hasContentIssue false
  • English
  • Français

Lundberg bounds on the tails of compound distributions

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Gordon E. Willmot  and
Xiaodong Lin
Gordon E. Willmot
Affiliation:
University of Waterloo
Xiaodong Lin*
Affiliation:
University of Waterloo
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1.
Get access Rights & Permissions [Opens in a new window]

Abstract

Exponential bounds are derived for the tail probabilities of various compound distributions, generalizing the classical Lundberg inequality of insurance risk theory. Failure rate properties of the compounding distribution including log-convexity and log-concavity are considered in some detail. Mixed Poisson compounding distributions are also considered. A ruin theoretic generalization of the Lundberg inequality is obtained in the case where the number of claims process is a mixed Poisson process. An application to the M/G/1 queue length distribution is given.

Keywords

INSURANCE RISKCOMPLETELY MONOTONELOG-CONVEXLOG-CONCAVECOMPOUND GEOMETRICMIXED POISSONFAILURE RATEMEAN RESIDUAL LIFETIMEM/G/1 QUEUE

MSC classification

Primary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 3 , September 1994 , pp. 743 - 756
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asmussen, S. (1987) Applied Probability and Queues. Wiley, Chichester.Google Scholar
Barlow, R. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Block, H. and Savits, T. (1980) Laplace transforms for classes of life distributions. Ann. Prob. 8, 465–474.Google Scholar
Eade, J. (1983) The ruin problem for mixed Poisson risk processes. Scandinavian Actuarial J. 193–210.Google Scholar
Esary, J., Marshall, A. and Proschan, F. (1973) Shock models and wear processes. Ann. Prob. 1, 627–649.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn revised. Wiley, New York.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd ed. Wiley, New York.Google Scholar
Gerber, H. (1979) An Introduction to Mathematical Risk Theory. S. S. Huebner Foundation, University of Pennsylvania, Philadelphia.Google Scholar
Grandell, J. (1993) The mixed Poisson process in a modern setting. Unpublished manuscript.Google Scholar
Grandell, J. and Peiram, L. (1973) A note on the ruin problem for a class of stochastic processes with interchangeable increments. Astin Bull. 7, 81–89.CrossRefGoogle Scholar
Johnson, N., Kotz, S. and Kemp, A. (1992) Univariate Discrete Distributions, 2nd ed. Wiley, New York.Google Scholar
Karlin, S. (1968) Total Positivity, Vol. 1. Stanford University Press.Google Scholar
Tijms, H. (1986) Stochastic Modelling and Analysis: A Computational Approach. Wiley, Chichester.Google Scholar
Van, Harn. K. (1978) Classifying Infinitely Divisible Distributions by Functional Equations. Math. Centre Tracts 103, Math. Centre, Amsterdam.Google Scholar