A decomposition for someU-type statistics |
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Authors: | Henry Teicher Cun-Hui Zhang |
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Institution: | 1. Department of Statistics, Rutgers University, 08903, New Brunswick, New Jersey
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Abstract: | Define , $S_{k,n} = \Sigma _{1 \leqslant i_1< \cdot \cdot \cdot< l_k \leqslant n} X_{i_1 } \cdot \cdot \cdot X_{i_k } ,n \geqslant k \geqslant {\text{1}}$ where {X, X n ,n≥1} are i.i.d. random variables withEX=0,EX 2=1 and letH k (·) denote the Hermite polynomial of degreek. By establishing an LIL for products of correlated sums of i.i.d. random variables, the a.s. decomposition $$\begin{gathered} k!S_{k,n} = n^{k/2} H_k (S_{1n} /n^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ) - \left( {\begin{array}{*{20}c} k \\ 2 \\ \end{array} } \right)S_{1.n}^{k - 2} \sum\limits_{i = 1}^n {(X_i^2 - 1)} \hfill \\ + O(n^{(k - 1)/2} (\log \log n)^{(k - 3/2} ) \hfill \\ \end{gathered} $$ valid whenEX 4<∞, elicits an LIL forη k,n =k!S k,n ?n k/2 H k (S 1n /n 1/2) under a reduced normalization. Moreover, whenE|X| p <∞ for somep in 2, 4], a Marcinkiewicz-Zygmund type strong law forη k,n is obtained, likewise under a reduced normalization. |
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