The Annals of Probability

On Distributions Related to Transitive Closures of Random Finite Mappings

Boris Pittel

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For $f$, a random single-valued mapping of an $n$-element set $X$ into itself, let $f^{-1}$ be the inverse mapping, and $f^\ast$ be such that $f^\ast(x) = \{f(x)\} \cup f^{-1}(x), x \in X$. For a given subset $A \subset X$, introduce three random variables $\xi(A) = |\hat f(A)|, \eta(A) = |\hat f^{-1}(A)|$, and $\zeta(A) = |\hat f^\ast(A)|$, where $\hat f, \hat f^{-1}, \hat f^\ast$ stand for transitive closures of $f, f^{-1}, f^\ast$. The distributions of $\xi(A)$ and $\zeta(A)$ are obtained. ($\eta(A)$ was earlier studied by J. D. Burtin.) For large $n$, the asymptotic behavior of those distributions is studied under various assumptions concerning $m = |A|$. For instance, it is shown that $\xi(A)$ is asymptotically normal with mean $(2mn)^{1/2}$ and variance $n/2$, and $(n - \zeta(A))(n/m)^{-1}$ is asymptotically $\mathscr{U}^2/2$ ($\mathscr{U}$ being the standard normal variable), provided $m \rightarrow \infty, m = o(n)$. The results are interpreted in terms of epidemic processes on random graphs introduced by I. Gertsbakh.

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Ann. Probab., Volume 11, Number 2 (1983), 428-441.

First available in Project Euclid: 19 April 2007

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Primary: 60C05: Combinatorial probability
Secondary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 05C30: Enumeration in graph theory

Random mappings and graphs enumeration generating functions distributions limit laws epidemic processes


Pittel, Boris. On Distributions Related to Transitive Closures of Random Finite Mappings. Ann. Probab. 11 (1983), no. 2, 428--441. doi:10.1214/aop/1176993608.

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