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January, 1989 A Functional Central Limit Theorem for Random Mappings
Jennie C. Hansen
Ann. Probab. 17(1): 317-332 (January, 1989). DOI: 10.1214/aop/1176991511


We consider the set of mappings of the integers $\{1, 2, \ldots, n\}$ into $\{1, 2, \ldots, n\}$ and put a uniform probability measure on this set. Any such mapping can be represented as a directed graph on $n$ labelled vertices. We study the component structure of the associated graphs as $n \rightarrow \infty$. To each mapping we associate a step function on $\lbrack 0, 1 \rbrack$. Each jump in the function equals the number of connected components of a certain size in the graph which represents the map. We normalize these functions and show that the induced measures on $D\lbrack 0, 1 \rbrack$ converge to Wiener measure. This result complements another result by Aldous on random mappings.


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Jennie C. Hansen. "A Functional Central Limit Theorem for Random Mappings." Ann. Probab. 17 (1) 317 - 332, January, 1989.


Published: January, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0667.60009
MathSciNet: MR972788
Digital Object Identifier: 10.1214/aop/1176991511

Primary: 60C05
Secondary: 60B10

Keywords: component structure , digraphs , Random graphs , Random mappings , Wiener measure

Rights: Copyright © 1989 Institute of Mathematical Statistics

Vol.17 • No. 1 • January, 1989
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