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 X1, . . . , Xn of a general Harris chain X may be divided into asymptotically i.i.d. data blocks B1, . . . , BN of random length corresponding to successive (pseudo-) regeneration times, we introduce the notion of regenerative U -statistic ΩN = ∑ k 6=l ωh(Bk,Bl)/(N(N − 1)) related to a U -statistic Un = ∑ i 6=j h(Xi, Xj)/(n(n − 1)). We show that, under mild conditions, these two statistics are asymptotically equivalent up to the order OP(n). This result serves as a basis for establishing limit theorems related to statistics of the same form as Un. 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.