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

Davide Giraudo

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.

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

Abstract

In this note, we give sufficient conditions for the almost sure and the convergence in of a -statistic of order built on a strictly stationary but not necessarily ergodic sequence.
Paper Structure (11 sections, 8 theorems, 64 equations)

This paper contains 11 sections, 8 theorems, 64 equations.

Table of Contents

  1. Introduction and main results
  2. Almost sure convergence
  3. Convergence in $\mathbb L^p$, $p\geqslant 1$
  4. Examples
  5. Proofs
  6. Proof of Theorem \ref{['thm:ergodic_theorem_sym_ps']}
  7. Proof of Corollary \ref{['cor:conv_Lp_sym']}
  8. Convergence of weighted averages
  9. Proof of Theorem \ref{['thm:conv_Lp_ergodique']}
  10. Proof of Theorem \ref{['thm:conv_Lp_non-ergodique_densite']}
  11. Proof of the results of Subsection \ref{['subsec:examples']}

Key Result

Theorem 1.1

Let $\left(S,d\right)$ be a separable metric space, let $\left(X_i\right)_{i\in\mathbb Z}=\left(X_0\circ T^i\right)_{i\in\mathbb Z}$ be a strictly stationary sequence. Suppose that $h\colon S^m\to\mathbb R$ satisfies the following assumptions: Then for almost every $\omega\in\Omega$, the following convergence holds: where $I_m\left(S,h,\omega\right)$ is defined as in eq:definition_of_I_h_omega.

Theorems & Definitions (12)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Lemma 2.1
  • proof
  • proof : Proof of Corollary \ref{['cor:kernel_test_symmetry']}
  • ...and 2 more