## The Annals of Probability

- Ann. Probab.
- Volume 17, Number 1 (1989), 317-332.

### A Functional Central Limit Theorem for Random Mappings

#### Abstract

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.

#### Article information

**Source**

Ann. Probab., Volume 17, Number 1 (1989), 317-332.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991511

**Digital Object Identifier**

doi:10.1214/aop/1176991511

**Mathematical Reviews number (MathSciNet)**

MR972788

**Zentralblatt MATH identifier**

0667.60009

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60C05: Combinatorial probability

Secondary: 60B10: Convergence of probability measures

**Keywords**

Random mappings random graphs digraphs Wiener measure component structure

#### Citation

Hansen, Jennie C. A Functional Central Limit Theorem for Random Mappings. Ann. Probab. 17 (1989), no. 1, 317--332. doi:10.1214/aop/1176991511. https://projecteuclid.org/euclid.aop/1176991511

#### Corrections

- See Correction: Jennie C. Hansen. Correction: A Functional Central Limit Theorem for Random Mappings. Ann. Probab., Volume 19, Number 3 (1991), 1393--1396.Project Euclid: euclid.aop/1176990350