The LIL for canonical U-statistics of order 2

Details The LIL for canonical U-statistics of order 2 The LIL for canonical U-statistics of order 2

1999-05-20

arxiv.org/abs/math/9905132v1

Mathematics

Probability

36 pages

Scientific paper

Let X,X_1,X_2,... 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, $\limsup_n (n\log\log n)^{-1}|\sum_{1<= i< j<= n}h(X_i,X_j)|<\infty$ a.s., holds if and only if the following three conditions are satisfied: h is canonical for the law of X (that is Eh(X,y)=0 for almost y) and there exists $C<\infty$ such that, both, $E\min(h^2(X_1,X_2),u)

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