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Functionals of random mappings: exact and asymptotic results

Published online by Cambridge University Press:  01 July 2016

P. J. Donnelly ,
W. J. Ewens  and
S. Padmadisastra
P. J. Donnelly*
Affiliation:
Queen Mary and Westfield College, London
W. J. Ewens*
Affiliation:
Monash University
S. Padmadisastra*
Affiliation:
Monash University
*
∗Postal address: School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London El 4NS, UK.
∗∗Postal address: Department of Mathematics, Monash University, Clayton, VIC 3168, Australia.
∗∗Postal address: Department of Mathematics, Monash University, Clayton, VIC 3168, Australia.
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Abstract

A random mapping partitions the set {1, 2, ···, m} into components, where i and j are in the same component if some functional iterate of i equals some functional iterate of j. We consider various functionals of these partitions and of samples from it, including the number of components of ‘small' size and of size O(m) as m → ∞the size of the largest component, the number of components, and various symmetric functionals of the normalized component sizes. In many cases exact results, while available, are uniformative, and we consider various approximations. Numerical and simulation results are also presented. A central tool for many calculations is the ‘frequency spectrum', both exact and asymptotic.

Keywords

FREQUENCY SPECTRUMSIZE-BIASED DISTRIBUTIONGEM DISTRIBUTION
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 3 , September 1991 , pp. 437 - 455
Copyright
Copyright © Applied Probability Trust 1991 

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