Abstract
Let be a -statistic based on a symmetric kernel and i.i.d. samples . In this paper, the exact moment convergence rates in the law of the iterated logarithm and the law of the logarithm of are obtained, which extend previous results concerning partial sums.
Highlights
Introduction and Main ResultLet h x, y be a real-valued Borel measurable function, symmetric in its arguments
Since Theorem A requires a strong condition, that is, E|Rn|2 < ∞, Yan and Su 7 investigated the precise asymptotics of U-statistics under minimal conditions and got the following result
Note that in our theorem, we assume E|h X1, X2 |2 < ∞, which is stronger than the condition imposed by Yan and Su 7, and required only to use a moment bound of Chen 9 given in Lemma 2.1
Summary
Introduction and Main ResultLet h x, y be a real-valued Borel measurable function, symmetric in its arguments. Let {Xn; n ≥ 1} be a sequence of i.i.d. random variables with mean zero and variance one. Since Theorem A requires a strong condition, that is, E|Rn|2 < ∞, Yan and Su 7 investigated the precise asymptotics of U-statistics under minimal conditions and got the following result.
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