A renewal approach to Markovian U-statistics

Abstract

In this paper we describe a novel approach to the study of U-statistics in the Markovian setup based on the (pseudo-) regenerative properties of Harris Markov chains. Exploiting the fact that any sample path X 1, ..., X n of a general Harris chain X may be divided into asymptotically i.i.d. data blocks \(\mathcal{B}_1 , \ldots \mathcal{B}_N\) of random length corresponding to successive (pseudo-) regeneration times, we introduce the notion of regenerative U-statistic Ω N = Σ k≠l ω h \(\left( {\mathcal{B}_k ,\mathcal{B}_l } \right)\)/(N(N − 1)) related to a U-statistic U n = Σ i≠j h(X i , X j )/(n(n − 1)). We show that, under mild conditions, these two statistics are asymptotically equivalent up to the order O ℙ(n−1). This result serves as a basis for establishing limit theorems related to statistics of the same form as U n . Beyond its use as a technical tool for proving results of a theoretical nature, the regenerative method is also employed here in a constructive fashion for estimating the limiting variance or the sampling distribution of certain U-statistics through resampling. The proof of the asymptotic validity of this statistical methodology is provided, together with an illustrative simulation result.