Asymptotics for products of sums and \(U\)-statistics
type: Item
instance of: scholarly article
title: Asymptotics for products of sums and \(U\)-statistics
zbMATH Open document ID: 1008.62048
author: Jacek Wesołowski
author: Grzegorz A. Rempala
published in: Electronic Communications in Probability
publication date: 2003-02-25T00:00:00Z
full work available at URL: https://eudml.org/doc/122501
full work available at URL: paper5.abs.html
review text: \textit{B.C. Arnold} and \textit{J.A. Villaseñor} [Extremes 1, No. 3, 351-363 (1999; Zbl 0931.60011)] established the asymptotic distribution for a product of partial sums of independent exponential random variables. This paper shows that the result is more universal and applies to products of partial sums of any positive, square integrable random variables. Specifically, let \(X_1, \ldots, X_n\) be independent and identically distributed positive random variables with \(EX_1 = \mu\) and Var\((X_1) = \sigma^2 < \infty.\) Let \(S_k = X_1 + \ldots + X_k.\) Then \([\prod_{k=1}^{n} (S_k/(k\mu))]^{\mu/(\sigma \sqrt{n})}\) converges in distribution to \(\exp(\sqrt{2} {\mathcal N})\) where \({\mathcal N}\) is a standard normal variable. An extension to products of \(U\)-statistics is provided.
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 1873756
zbMATH Keywords: central limit theorem
zbMATH Keywords: lognormal distribution
zbMATH Keywords: products of sums
zbMATH Keywords: U-statistics
reviewed by: Neville Weber
MaRDI profile type: Publication
Wikidata QID: Q56070031
OpenAlex ID: W2042382772
DOI: 10.1214/ECP.V7-1046
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description: scientific article; zbMATH DE number 1873756