Asymptotics for products of sums and \(U\)-statistics
\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.
scientific article; zbMATH DE number 1873756