Limit theorems for U-statistics of Bernoulli data

Abstract

In this paper, we consider U -statistics whose data is a strictly stationary sequence which can be expressed as a functional of an i.i.d. one. We establish a strong law of large numbers, a bounded law of the iterated logarithms and a central limit theorem under a dependence condition. The main ingredients for the proof are an approximation by U -statistics whose data is a functional of i.i.d. random variables and an analogue of the Hoeffding's decomposition for U -statistics of this type.

Deviation inequalities for Banach space valued martingales differences sequences and random fields

“…Such a result was known when m = 2 and B = R (see Proposition 1.2 in [21]). Notice that we do not need symmetry of the kernel h. Note that this is not a direct application of the established deviation inequality.…”

U-statistics of local sample moments under weak dependence

“…We now state a central limit theorem for U-statistics of this more concrete type of triangular array (1.2). The question of the central limit theorem for U-statistics whose entries are Bernoulli shifts has been addressed in Hsing and Wu [2004] and Giraudo [2021], but these results do not treat the case of arrays.…”