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Lundberg inequalities for renewal equations

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Gordon E. Willmot ,
Jun Cai  and
X. Sheldon Lin
Gordon E. Willmot*
Affiliation:
University of Waterloo
Jun Cai*
Affiliation:
University of Waterloo
X. Sheldon Lin*
Affiliation:
University of Toronto
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. Email address: gewillmo@uwaterloo.ca
∗∗ Current address: Centre for Actuarial Studies, Faculty of Economics and Commerce, University of Melbourne, Victoria 3010, Australia.
∗∗∗ Postal address: Department of Statistics, University of Toronto, Toronto, Ontario, Canada M5S 3G3.
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Abstract

Sharp upper and lower bounds are derived for the solution of renewal equations. These include as special cases exponential inequalities, some of which have been derived for specific renewal equations. Together with the well-known Cramér-Lundberg asymptotic estimate, these bounds give additional information about the behaviour of the solution. Nonexponential bounds, which are of use in connection with defective renewal equations, are also obtained. The results are then applied in examples involving the severity of insurance ruin, age-dependent branching processes, and a generalized type II Geiger counter.

Keywords

Adjustment coefficientdefective renewal equationscompound geometricCramér-Lundberg asymptotic estimateseverity of ruinbranching process

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 3 , September 2001 , pp. 674 - 689
Copyright
Copyright © Applied Probability Trust 2001 

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