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A renewal approach to Markovian \(U\)-statistics. (English) Zbl 1308.62162

Summary: 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 \(X_1,\dots,X_n\) of a general Harris chain \(X\) may be divided into asymptotically i.i.d. data blocks \(\mathcal B_1,\dots,\mathcal B_N\) of random length corresponding to successive (pseudo-) regeneration times, we introduce the notion of regenerative \(U\)-statistic \(\Omega_N=\sum_{k\neq l}\omega_h(\mathcal B_k,\mathcal B_l/(N(N-1))\) related to a \(U\)-statistic \(U_n=\sum_{i\neq j}h(X_i,X_j)/(n(n-1))\). We show that, under mild conditions, these two statistics are asymptotically equivalent up to the order \(O_{\mathbb P}(n-1)\). 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.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62F10 Point estimation
62F12 Asymptotic properties of parametric estimators
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