A functional central limit theorem for random mappings. (English) Zbl 0667.60009

Hansen, Jennie C.

Ann. Probab. 17, No. 1, 317-332 (1989).

Let \(T_ n\) denote the set of all single-valued mappings from \(\{\) 1,2,...,n\(\}\) into itself. A uniform probability measure \(P_ n\) is defined on \(T_ n\) by \(P_ n(\phi)=1/n^ n\) for each \(\phi \in T_ n\). It is well-known that each \(\phi \in T_ n\) may be represented by a directed graph \(G_{\phi}\). Let \(X_ n(t,\phi)\) denote the number of connected components in \(G_{\phi}\) with no more than \(n^ t\) vertices, where \(0\leq t\leq 1\), and let \[ Y_ n(t,\phi)=(X_ n(t,\phi)-(t/2)\log n)/((1/2)\log n)^{1/2}. \] The author proves that the sequence of induced measures \(P_ n\circ Y_ n^{-1}\) converges weakly to the Wiener measure W on (D[0,1],\({\mathcal D})\) as \(n\to \infty\), where D[0,1] and \({\mathcal D}\) denote the space of right-continuous functions with left limits on [0,1] and the \(\sigma\)-algebra generated by the Borel sets of D[0,1] with respect to the Skorokhod topology on D[0,1], respectively.

Reviewer: L.Mutafchiev

MSC:

60C05	Combinatorial probability
60B10	Convergence of probability measures
60F17	Functional limit theorems; invariance principles

Keywords:

random mappings; Wiener measure; Skorokhod topology