A Limit Result for U-Statistics of Binary Variables

Sergey Utev and Niels G. Becker
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Sergey Utev: La Trobe University
Niels G. Becker: La Trobe University

Journal of Theoretical Probability, 1998, vol. 11, issue 3, 853-856

Abstract: 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(ċ) is the Hermite polynomial of degree k, and {X, X n, n≥1} are independent identically distributed binary random variables with Pr(Xε{−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 EX≠0, respectively.

Keywords: U-type statistics; binary random variables; law of the iterated logarithm (search for similar items in EconPapers)
Date: 1998
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DOI: 10.1023/A:1022666901529

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