A note on prediction for discrete time series

Morvai, Gusztáv ; Weiss, Benjamin

Kybernetika, Tome 48 (2012) no. 4, pp. 809-823.

Résumé

Let $\{X_n\}$ be a stationary and ergodic time series taking values from a finite or countably infinite set ${\cal X}$ and that $f(X)$ is a function of the process with finite second moment. Assume that the distribution of the process is otherwise unknown. We construct a sequence of stopping times $\lambda_n$ along which we will be able to estimate the conditional expectation $E(f(X_{\lambda_n+1})|X_0,\dots,X_{\lambda_n} )$ from the observations $(X_0,\dots,X_{\lambda_n})$ in a point wise 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 (in particular, all stationary and ergodic Markov chains are included in this class) then $ \lim_{n\to \infty} \frac{n}{\lambda_n}>0$ almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then $\lambda_n$ is upper bounded by a polynomial, eventually almost surely.

Détail

Classification : 60G10, 60G25, 62G05
Keywords: nonparametric estimation; stationary processes