Random permutations and Brownian motion
type: Item
instance of: scholarly article
title: Random permutations and Brownian motion
zbMATH Open document ID: 0578.60033
author: J. M. DeLaurentis
author: Boris Pittel
published in: Pacific Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60C05
zbMATH DE Number: 3925889
zbMATH Keywords: cycles of permutations
zbMATH Keywords: uniform distribution
reviewed by: August Michał Zapała
MaRDI profile type: Publication
full work available at URL: pjm.1985.119.287
OpenAlex ID: W2053390290
Wikidata QID: Q104786192
DOI: 10.2140/PJM.1985.119.287
description: scientific article; zbMATH DE number 3925889