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On the occurrence of composite events and clusters of points

Part of: Distribution theory Multivariate analysis Special processes Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  14 July 2016

Valeri T. Stefanov
Valeri T. Stefanov*
Affiliation:
The University of Western Australia
*
∗Postal address: Department of Mathematics and Statistics, The University of Western Australia, Nedlands, WA 6907, Australia. Email address: stefanov@maths.uwa.edu.au.
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Abstract

We derive explicit closed expressions for the moment generating functions of whole collections of quantities associated with the waiting time till the occurrence of composite events in either discrete or continuous-time models. The discrete-time models are independent, or Markov-dependent, binary trials and the events of interest are collections of successes with the property that each two consecutive successes are separated by no more than a fixed number of failures. The continuous-time models are renewal processes and the relevant events are clusters of points. We provide a unifying technology for treating both the discrete and continuous-time cases. This is based on first embedding the problems into similar ones for suitably selected Markov chains or Markov renewal processes, and second, applying tools from the exponential family technology.

Keywords

Markov chainstopping timerunpoint processcluster

MSC classification

Primary: 60E10: Characteristic functions; other transforms 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)
Secondary: 62E15: Exact distribution theory 62H10: Distribution of statistics
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 1012 - 1018
Copyright
Copyright © Applied Probability Trust 1999 

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