Some notes on ergodic theorem for $U$-statistics of order $m$ for stationary and not necessarily ergodic sequences

TL;DR

This paper establishes conditions under which $U$-statistics of order $m$ converge almost surely and in $ ext{L}^p$ for stationary sequences that may not be ergodic, extending classical results.

Contribution

It provides new sufficient conditions for convergence of $U$-statistics in non-ergodic stationary sequences, broadening the applicability of ergodic theorems.

Findings

Sufficient conditions for almost sure convergence

Conditions for convergence in $ ext{L}^p$

Extension to non-ergodic stationary sequences

Abstract

In this note, we give sufficient conditions for the almost sure and the convergence in $\mathbb{L}^p$ of a $U$-statistic of order $m$ built on a strictly stationary but not necessarily ergodic sequence.