Roussas, George G. 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 Cited in 10 Documents MSC: 62M05 Markov processes: estimation; hidden Markov models 62G20 Asymptotic properties of nonparametric inference 62E20 Asymptotic distribution theory in statistics Keywords:quantiles; strong consistency; asymptotic normality; strictly stationary, discrete-time Markov process; one-step transition probability distribution; ratio of kernel estimates; rho mixing Citations:Zbl 0722.00032 PDFBibTeX XMLCite \textit{G. G. Roussas}, in: Laws of the iterated logarithm for density estimators. . 443--462 (1991; Zbl 0735.62081)