# Limit theorems for U-statistics of Bernoulli data

## TL;DR

This paper establishes limit theorems such as the law of large numbers, iterated logarithm, and central limit theorem for U-statistics derived from stationary Bernoulli data modeled as functionals of i.i.d. sequences, under dependence conditions.

## Contribution

It introduces new limit theorems for U-statistics of dependent Bernoulli data, extending classical results to dependent sequences with a novel approach.

## Key findings

- Proves strong law of large numbers for these U-statistics.
- Establishes a bounded law of the iterated logarithm.
- Derives a central limit theorem under dependence conditions.

## 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 $\ell$ i.i.d. random variables and an analogue of the Hoeffding's decomposition for U-statistics of this type.