On Distributions Related to Transitive Closures of Random Finite Mappings

Abstract

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.