A note on prediction for discrete time series

Morvai, Gusztáv, and Weiss, Benjamin. "A note on prediction for discrete time series." Kybernetika 48.4 (2012): 809-823. <http://eudml.org/doc/246722>.

@article{Morvai2012,
abstract = {Let $\lbrace X_n\rbrace $ be a stationary and ergodic time series taking values from a finite or countably infinite set $\{\mathcal \{X\}\}$ 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\rightarrow \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.},
author = {Morvai, Gusztáv, Weiss, Benjamin},
journal = {Kybernetika},
keywords = {nonparametric estimation; stationary processes; stationary processes; nonparametric estimation},
language = {eng},
number = {4},
pages = {809-823},
publisher = {Institute of Information Theory and Automation AS CR},
title = {A note on prediction for discrete time series},
url = {http://eudml.org/doc/246722},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Morvai, Gusztáv
AU - Weiss, Benjamin
TI - A note on prediction for discrete time series
JO - Kybernetika
PY - 2012
PB - Institute of Information Theory and Automation AS CR
VL - 48
IS - 4
SP - 809
EP - 823
AB - Let $\lbrace X_n\rbrace $ be a stationary and ergodic time series taking values from a finite or countably infinite set ${\mathcal {X}}$ 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\rightarrow \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.
LA - eng
KW - nonparametric estimation; stationary processes; stationary processes; nonparametric estimation
UR - http://eudml.org/doc/246722
ER -