The Annals of Probability

The LIL for canonical U-statistics of order 2

Evarist Giné, Stanislaw Kwapień, Rafał Latała, and Joel Zinn

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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 $\sup\{Eh(X_1,X_2) \times f(X_1)g(X_2) : ||f(X)||_2 \le 1, ||g(X)||_2 \le 1; ||f||_\infty < \infty, ||g||_\infty < \infty\} \le C$.

Article information

Ann. Probab., Volume 29, Number 1 (2001), 520-557.

First available in Project Euclid: 21 December 2001

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F15: Strong theorems

U-statistics (canonical or degenerate) law of the iterated logarithm


Giné, Evarist; Kwapień, Stanislaw; Latała, Rafał; Zinn, Joel. The LIL for canonical U -statistics of order 2. Ann. Probab. 29 (2001), no. 1, 520--557. doi:10.1214/aop/1008956343.

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