A renewal approach to Markovian U-statistics

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 B ₁,...B _n of random length corresponding to successive (pseudo-) regeneration times, we introduce the notion of regenerative U-statistic Ω _N = Σ _k≠lω _h(B _k,B _l)/(N(N - 1)) related to a U-statistic U _n = Σ _i≠jh(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.