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On higher-order properties of compound geometric distributions

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Gordon E. Willmot
Gordon E. Willmot*
Affiliation:
University of Waterloo
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. Email address: gewillmo@uwaterloo.ca
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Abstract

An explicit convolution representation for the equilibrium residual lifetime distribution of compound zero-modified geometric distributions is derived. Second-order reliability properties are seen to be essentially preserved under geometric compounding, and complement results of Brown (1990) and Cai and Kalashnikov (2000). The convolution representation is then extended to the nth-order equilibrium distribution. This higher-order convolution representation is used to evaluate the stop-loss premium and higher stop-loss moments of the compound zero-modified geometric distribution, as well as the Laplace transform of the kth moment of the time of ruin in the classical risk model.

Keywords

Compound geometric convolutionresidual lifetime distributionequilibrium distributionintegrated taildefective renewal equationDFRIFRIMRLDMRLNWUNBU2-NWU2-NBUNWUCNBUCNWUENBUEmean residual lifetimeLaplace-Stieltjes transformstop-loss premiumstop-loss momentsLundberg boundtime of ruin

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 2 , June 2002 , pp. 324 - 340
Copyright
Copyright © Applied Probability Trust 2002 

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