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Intermittent estimation of stationary time series. (English) Zbl 1082.62073
Summary: Let $$\{X_n\}^\infty_{n=0}$$ be a stationary real-valued time series with unknown distribution. Our goal is to estimate the conditional expectation of $$X_{n+1}$$ based on the observations $$X_i$$, $$0\leq i\leq n$$, in a strongly consistent way. D. Bailey [Sequential schemes for classifying and predicting ergodic processes. Ph.D. thesis, Stanford Univ. (1976)] and B. Ya. Ryabko [Probl. Inf. Transm. 24, 87–96 (1988), translation from Probl. Peredachi Inf. 24, 3–14 (1988; Zbl 0666.94009)] proved that this is not possible even for ergodic binary time series if one estimates at all values of $$n$$. We propose a very simple algorithm which will make predictions infinitely often at carefully selected stopping times chosen by our rule. We show that under certain conditions our procedure is strongly (pointwise) consistent, and $$L_2$$ consistent without any condition. An upper bound on the growth of the stopping times is also presented in this paper.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62G05 Nonparametric estimation 62M20 Inference from stochastic processes and prediction 60G25 Prediction theory (aspects of stochastic processes) 60G10 Stationary stochastic processes
Zbl 0666.94009
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