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Shocks, runs and random sums

Part of: Operations research and management science Special processes Limit theorems Real functions Stochastic processes

Published online by Cambridge University Press:  14 July 2016

F. Mallor  and
E. Omey
F. Mallor*
Affiliation:
Universidad Pública de Navarra
E. Omey*
Affiliation:
Economische Hogeschool Sint-Aloysius
*
∗ Postal address: Department of Statistics and Operations Research, Universidad Pública de Navarra, Campus Arrosadia, 31006 Pamplona, Spain. Email address: mallor@unavarra.es
∗∗ Postal address: Department of Mathematics and Statistics, Economische Hogeschool Sint-Aloysius, Stormstraat 2, 1000-Brussels, Belgium.
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Abstract

In this paper we study random variables related to a shock reliability model. Our models can be used to study systems that fail when k consecutive shocks with critical magnitude (e.g. above or below a certain critical level) occur. We obtain properties of the distribution function of the random variables involved and we obtain their limit behaviour when k tends to infinity or when the probability of entering a critical set tends to zero. This model generalises the Poisson shock model.

Keywords

shock modelrandom sumsreliabilityregular variation

MSC classification

Primary: 60F05: Central limit and other weak theorems 60K10: Applications (reliability, demand theory, etc.) 90B25: Reliability, availability, maintenance, inspection
Secondary: 26A12: Rate of growth of functions, orders of infinity, slowly varying functions 60G50: Sums of independent random variables; random walks
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 2 , June 2001 , pp. 438 - 448
Copyright
Copyright © Applied Probability Trust 2001 

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