### Abstract

Let {X _{n}} be a stationary and ergodic time series taking values from a finite or countably infinite set 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 λ _{n} along which we will be able to estimate the conditional expectation E(f(X _{λn+1})|X _{0}, . . . ,X _{λn}) from the observations (X _{0}, . . . ,X _{λn}) 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 limn→ ∞n/λ _{n} > 0 almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then λ _{n} is upper bounded by a polynomial, eventually almost surely.

Original language | English |
---|---|

Pages (from-to) | 809-823 |

Number of pages | 15 |

Journal | Kybernetika |

Volume | 48 |

Issue number | 4 |

Publication status | Published - 2012 |

### Fingerprint

### Keywords

- Nonparametric estimation
- Stationary processes

### ASJC Scopus subject areas

- Software
- Artificial Intelligence
- Control and Systems Engineering
- Information Systems
- Theoretical Computer Science
- Electrical and Electronic Engineering

### Cite this

*Kybernetika*,

*48*(4), 809-823.

**A note on prediction for discrete time series.** / Morvai, G.; Weiss, Benjamin.

Research output: Contribution to journal › Article

*Kybernetika*, vol. 48, no. 4, pp. 809-823.

}

TY - JOUR

T1 - A note on prediction for discrete time series

AU - Morvai, G.

AU - Weiss, Benjamin

PY - 2012

Y1 - 2012

N2 - Let {X n} be a stationary and ergodic time series taking values from a finite or countably infinite set 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 λ n along which we will be able to estimate the conditional expectation E(f(X λn+1)|X 0, . . . ,X λn) from the observations (X 0, . . . ,X λn) 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 limn→ ∞n/λ n > 0 almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then λ n is upper bounded by a polynomial, eventually almost surely.

AB - Let {X n} be a stationary and ergodic time series taking values from a finite or countably infinite set 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 λ n along which we will be able to estimate the conditional expectation E(f(X λn+1)|X 0, . . . ,X λn) from the observations (X 0, . . . ,X λn) 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 limn→ ∞n/λ n > 0 almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then λ n is upper bounded by a polynomial, eventually almost surely.

KW - Nonparametric estimation

KW - Stationary processes

UR - http://www.scopus.com/inward/record.url?scp=84866022828&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84866022828&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84866022828

VL - 48

SP - 809

EP - 823

JO - Kybernetika

JF - Kybernetika

SN - 0023-5954

IS - 4

ER -