A simple variance inequality for U-statistics of a Markov chain with applications

Gersende Fort (1) , Eric Moulines (1) , Pierre Priouret (2) , Pierre Vandekerkhove (3)

1 LTCI - Laboratoire Traitement et Communication de l'Information
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
3 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

We establish a simple variance inequality for U-statistics whose underlying sequence of random variables is an ergodic Markov Chain. The constants in this inequality are explicit and depend on computable bounds on the mixing rate of the Markov Chain. We apply this result to derive the strong law of large number for U-statistics of a Markov Chain under conditions which are close from being optimal.

Mots clés: U-statistics; Markov chains; Inequalities; Limit theorems; Law of large numbers

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Gersende Fort, Eric Moulines, Pierre Priouret, Pierre Vandekerkhove. A simple variance inequality for U-statistics of a Markov chain with applications. Statistics and Probability Letters, 2012, 82 (6), pp.1193--1201. ⟨hal-00608574⟩