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Estimation of transition distribution function and its quantiles in Markov processes: Strong consistency and asymptotic normality. (English) Zbl 0735.62081

Nonparametric functional estimation and related topics, NATO ASI Ser., Ser. C 335, 443-462 (1991).
[For the entire collection see Zbl 0722.00032.]
Based on the observation of the first \(n\) random variables \(X_ 1,\ldots,X_ n\) from a strictly stationary, discrete-time Markov process \(X\) the author investigates the asymptotic properties of a natural nonparametric estimate of the one-step transition probability distribution function as \(n\) tends to infinity. (The estimate is obtained by integrating the ratio of kernel estimates for the stationary two- and one-dimensional distributions.) Under appropriate conditions it is shown that this estimate is uniformly (in the main argument) strongly consistent and asymptotically normal; further, a natural estimate for the p-th quantile \((p\in(0,1))\) is shown to be strongly consistent.
Besides certain regularity assumptions a key assumption is that \(X\) has only one ergodic set, which has no cyclically moving subsets, and fulfils a \(\rho\)-mixing condition where \(\rho\) decreases of polynomal order.
[Editorial remark: See also the author’s article reviewed below.].
Reviewer: H.-M.Dietz

MSC:

62M05 Markov processes: estimation; hidden Markov models
62G20 Asymptotic properties of nonparametric inference
62E20 Asymptotic distribution theory in statistics

Citations:

Zbl 0722.00032
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