This article is cited in 23 papers

Convergence rates in the law of large numbers for Banach-valued dependent variables

J. Dedecker, F. Merlevede

Université Pierre & Marie Curie, Paris VI

Abstract: We extend Marcinkievicz–Zygmund strong laws of large numbers for martingales to weakly dependent random variables with values in smooth Banach spaces. The conditions are expressed in terms of conditional expectations. In the case of Hilbert spaces, we show that our conditions are weaker than optimal ones for strongly mixing sequences (which were previously known for real-valued variables only). As a consequence, we give rates of convergence for Cramér–von Mises statistics and for the empirical estimator of the covariance operator of a Hilbert-valued autoregressive process.

Keywords: smooth Banach spaces, Hilbert spaces, Marcinkievicz–Zygmund strong laws of large numbers, almost sure convergence, martingales, weak dependence, Cramér–von Mises statistics.

Received: 13.11.2003
Revised: 13.03.2006

Language: English

DOI: 10.4213/tvp78

 English version: Theory of Probability and its Applications, 2008, 52:3, 416–438

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