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Occupancy schemes associated to Yule processes

Part of: Markov processes Special processes

Published online by Cambridge University Press:  01 July 2016

Philippe Robert  and
Florian Simatos
Philippe Robert*
Affiliation:
INRIA
Florian Simatos*
Affiliation:
INRIA
*
∗ Postal address: INRIA Paris-Rocquencourt, Domaine de Voluceau, BP 105, 78153 Le Chesnay, France.
∗ Postal address: INRIA Paris-Rocquencourt, Domaine de Voluceau, BP 105, 78153 Le Chesnay, France.
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Abstract

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An occupancy problem with an infinite number of bins and a random probability vector for the locations of the balls is considered. The respective sizes of the bins are related to the split times of a Yule process. The asymptotic behavior of the landscape of the first empty bins, i.e. the set of corresponding indices represented by point processes, is analyzed and convergences in distribution to mixed Poisson processes are established. Additionally, the influence of the random environment, the random probability vector, is analyzed. It is represented by two main components: an independent, identically distributed sequence and a fixed random variable. Each of these components has a specific impact on the qualitative behavior of the stochastic model. It is shown in particular that, for some values of the parameters, some rare events, which are identified, determine the asymptotic behavior of the average values of the number of empty bins in some regions.

Keywords

Occupancy schemePoisson processconvergence of point processesrandom environmentrare events

MSC classification

Primary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)
Secondary: 60K37: Processes in random environments 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 2 , June 2009 , pp. 600 - 622
Copyright
Copyright © Applied Probability Trust 2009 

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