Random permutations and Brownian motion (Q1066544)

Let \(X_{ns}=X_{ns}(\sigma)\) denote the random number of cycles of length s in the permutation \(\sigma\) of a set \(\{\) 1,2,...,n\(\}\) selected randomly according to the uniform distribution and let \(Y_ n(t)=(\sum_{1\leq s\leq n^ t}X_{ns}-t \ln n)/\ln^{1/2} n\), \(0\leq t\leq 1\). The main theorem of the paper states that for each functional f: D\(<0,1>\to R\) continuous in the sup-norm metric, \(f(Y_ n)\) converges weakly to f(W), where W is the standard Brownian motion with continuous sample paths. As a corollary e.g. the \textit{P. Erdős} and \textit{P. Turan} theorem [Z. Wahrscheinlichkeitstheor. Verw. Geb. 4, 175-186 (1965; Zbl 0137.256)] is derived.