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Prediction for discrete time series. (English) Zbl 1061.62148

Summary: Let \(\{X_n\}\) be a stationary and ergodic time series taking values from a finite or countably infinite set \(\mathcal X\). Assume that the distribution of the process is otherwise unknown. We propose a sequence of stopping times \(\lambda_n\) along which we will be able to estimate the conditional probability \(P(X_{\lambda_n+1} = x \mid X_0,\ldots,X_{\lambda_n})\) from a data segment \((X_0,\dots,X_{\lambda_n})\) in a pointwise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series which includes, among others, all stationary and ergodic finitarily Markovian processes. If the stationary and ergodic process turns out to be finitarily Markovian (among others, all stationary and ergodic Markov chains are included in this class) then \(\lim_{n\to\infty} n/(\lambda_n) > 0\) almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then \(\lambda_n\) is upperbounded by a polynomial, eventually almost surely.

MSC:

62M20 Inference from stochastic processes and prediction
62G05 Nonparametric estimation
60G25 Prediction theory (aspects of stochastic processes)
60G10 Stationary stochastic processes
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