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February 2001 The LIL for canonical U-statistics of order 2
Evarist Giné, Stanislaw Kwapień, Rafał Latała, Joel Zinn
Ann. Probab. 29(1): 520-557 (February 2001). DOI: 10.1214/aop/1008956343


Let $X, X_i, i \in \mathbf{N}$, be independent identically distributed random variables and let $h(x,y) = h(y,x)$ be a measurable function of two variables. It is shown that the bounded law of the iterated logarithm, $\limsup _n \log n(\log n)^{-1}|\sum_{1\le i<j\le n} h(X_i, X_j)| < \infty$ a.s., holds if and only if the following three conditions are satisfied: $h$ is canonical for the law of $X$ [i.e., $Eh(X,y) = 0$ for almost all $y$] and there exists $C < \infty$ such that both $E(h^2(X_1,X_2) \wedge u) \le C \log \log u$ for all large $u$ and $\sup\{Eh(X_1,X_2) \times f(X_1)g(X_2) : ||f(X)||_2 \le 1, ||g(X)||_2 \le 1; ||f||_\infty < \infty, ||g||_\infty < \infty\} \le C$.


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Evarist Giné. Stanislaw Kwapień. Rafał Latała. Joel Zinn. "The LIL for canonical U-statistics of order 2." Ann. Probab. 29 (1) 520 - 557, February 2001.


Published: February 2001
First available in Project Euclid: 21 December 2001

zbMATH: 1014.60026
MathSciNet: MR1825163
Digital Object Identifier: 10.1214/aop/1008956343

Primary: 60F15

Keywords: Law of the iterated logarithm , U-statistics (canonical or degenerate)

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.29 • No. 1 • February 2001
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