A note on the ASCLT for triangular arrays of random variables with an extension to U-statistics
Let \(\{X_{ki}\}\) be a triangular array of independent and identically distributed positive random variables with finite absolute moments of order \(p>2.\) The authors extend the results of \textit{K. Gonchigdanzan} and \textit{G. Rempala} [Appl. Math. Lett. 19, No. 2, 191--196 (2006; Zbl 1088.60021)] to obtain an almost sure central limit theorem for products \(\prod_{k=1}^n S_k/k\), where \(S_k = \sum_{i=1}^{k} X_{ki}.\) The result is then applied to obtain the corresponding result for products of non-degenerate \(U\)-statistics with positive kernels based on a triangular array of iid random variables. Related triangular array results can be found by \textit{Y. Miao} [J. Dyn. Syst. Geom. Theor. 7, No. 1, 49--60 (2009; Zbl 1180.60030)].
scientific article; zbMATH DE number 6145424
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