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A Limit Result for U-Statistics of Binary Variables

Authors:	Sergey Utev Niels G Becker

Institution:	(1) School of Statistical Science, La Trobe University, Australia, 3083

Abstract:	Define $\eta _{k,n} = U_{k,n} - n^{k/2} H_2 (\sum\nolimits_{j = 1}^n {X_j /\sqrt n }$ , where $U_{k,n} = \sum\nolimits_{1 \leqslant i_1 \ne \cdots \ne i_k \leqslant n} {X_{i_1 } \cdots X_{i_k } }$ is a symmetric U-type statistic, H _k() is the Hermite polynomial of degree k, and {X, X _n, n1} are independent identically distributed binary random variables with Pr(X{–1, 1}})=1. We show that $\lim \sup \frac{{\eta _{k,n} }}{{(2nLLn)^{(k - 2)/2} }} = 2\left( {_3^k } \right){\text{ or }}\mathop {{\text{lim sup}}}\limits_{n \to \infty } \frac{{\eta _{k,n} }}{{(nEX)^{(k - 2)} }} = 2\left( {_3^k } \right){\text{ a}}{\text{.s}}{\text{.}}$ according as EX=0 or EX0, respectively.

Keywords:	U-type statistics binary random variables law of the iterated logarithm
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