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Convergence of the empirical two-sample $U$-statistics with $β$-mixing data

Herold Dehling, Davide Giraudo, Olimjon Sharipov

Abstract

We consider the empirical two-sample $U$-statistic with strictly $β$-mixing strictly stationary data and inverstigate its convergence in Skorohod spaces. We then provide an application of such convergence.

Convergence of the empirical two-sample $U$-statistics with $β$-mixing data

Abstract

We consider the empirical two-sample -statistic with strictly -mixing strictly stationary data and inverstigate its convergence in Skorohod spaces. We then provide an application of such convergence.

Paper Structure

This paper contains 16 sections, 17 theorems, 163 equations.

Table of Contents

  1. Introduction and main results
  2. Convergence of the two-sample $U$-statistic in Skorohod spaces $D\left([-R,R]\times [0,1]\right)$
  3. Application
  4. Proof
  5. Proof of Theorem \ref{['thm:convergence_two_sample_emp_Ustat']}
  6. Hoeffding's decomposition
  7. Convergence of the finite dimensional distributions of the linear part
  8. Convergence of the linear part
  9. Negligibility of the degenerated part
  10. Proof of Theorem \ref{['thm:convergence_two_sample_emp_Ustat_prob_replaced']}
  11. Hoeffding's decomposition
  12. Convergence of the finite dimensional distributions
  13. Convergence of the linear part
  14. Negligibility of the degenerated part
  15. Proof of Corollary \ref{['cor:application']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\left(X_i\right)_{i\in\mathbb Z}$ be a strictly stationary sequence. Let $e_n$ be the two-sample $U$-statistics empirical process with kernel $g\colon \mathbb R\times\mathbb R\to \mathbb R$ defined for $n\geqslant 1$, $0\leqslant t\leqslant 1$ and $s\in \mathbb R$ by Suppose that the following four conditions holds. Then for all $R$, where $\left(W\left(s,t\right), s\in \mathbb R,t\in [0,1

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Theorem 2.3
  • ...and 12 more