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Compound random mappings

Part of: Combinatorial probability Graph theory Limit theorems

Published online by Cambridge University Press:  14 July 2016

Jennie C. Hansen  and
Jerzy Jaworski
Jennie C. Hansen*
Affiliation:
Heriot-Watt University
Jerzy Jaworski*
Affiliation:
Adam Mickiewicz University
*
∗ Postal address: Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: j.hansen@ma.hw.ac.uk
∗∗ Postal address: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland.
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Abstract

In this paper, we introduce a compound random mapping model which can be viewed as a generalization of the basic random mapping model considered by Ross and by Jaworski. We investigate a particular example, the Poisson compound random mapping, and compare results for this model with results known for the well-studied uniform random mapping model. We show that, although the structure of the components of the random digraph associated with a Poisson compound mapping differs from the structure of the components of the random digraph associated with the uniform model, the limiting distribution of the normalized order statistics for the sizes of the components is the same as in the uniform case, i.e. the limiting distribution is the Poisson-Dirichlet (½) distribution on the simplex {{xi} : ∑ xi ≤ 1, xi ≥ xi+1 ≥ 0 for every i ≥ 1}.

Keywords

Random mappingsPoisson-Dirichlet distributioncomponent structure

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60F05: Central limit and other weak theorems 05C80: Random graphs
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 4 , December 2002 , pp. 712 - 729
Copyright
Copyright © Applied Probability Trust 2002 

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