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Tails of Stopped Random Products: The Factoid and Some Relatives

Part of: Markov processes Mathematical sociology Limit theorems Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Anthony G. Pakes
Anthony G. Pakes*
Affiliation:
University of Western Australia
*
∗ Postal address: School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia. Email address: pakes@maths.uwa.edu.au
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Abstract

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The upper tail behaviour is explored for a stopped random product ∏j=1 N X j , where the factors are positive and independent and identically distributed, and N is the first time one of the factors occupies a subset of the positive reals. This structure is motivated by a heavy-tailed analogue of the factorial n!, called the factoid of n. Properties of the factoid suggested by computer explorations are shown to be valid. Two topics about the determination of the Zipf exponent in the rank-size law for city sizes are discussed.

Keywords

Stopped random productheavy taillimit theoremrenewal theoryconvolution equivalenceZipf lawMarkov chaininvariant measure

MSC classification

Secondary: 60E05: Distributions 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks 60J05: Discrete-time Markov processes on general state spaces 91D10: Models of societies, social and urban evolution

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 4 , December 2008 , pp. 1161 - 1180
Copyright
Copyright © Applied Probability Trust 2008 

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