### Abstract

Let {X _{n} } be a stationary and ergodic time series taking values from a finite or countably infinite set X. Assume that the distribution of the process is otherwise unknown. We propose a sequence of stopping times λ _{n} along which we will be able to estimate the conditional probability P(Xλ _{n+1}=x|X _{0},...,λ _{n}) from data segment (X _{0},...,λ _{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→∞} n/λ _{n} > almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then λ _{n} is upperbounded by a polynomial, eventually almost surely.

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

Pages (from-to) | 1-12 |

Number of pages | 12 |

Journal | Probability Theory and Related Fields |

Volume | 132 |

Issue number | 1 |

DOIs | |

Publication status | Published - May 2005 |

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### Keywords

- Nonparametric estimation
- Stationary processes

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Statistics and Probability

### Cite this

*Probability Theory and Related Fields*,

*132*(1), 1-12. https://doi.org/10.1007/s00440-004-0386-3

**Prediction for discrete time series.** / Morvai, G.; Weiss, Benjamin.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 132, no. 1, pp. 1-12. https://doi.org/10.1007/s00440-004-0386-3

}

TY - JOUR

T1 - Prediction for discrete time series

AU - Morvai, G.

AU - Weiss, Benjamin

PY - 2005/5

Y1 - 2005/5

N2 - Let {X n } be a stationary and ergodic time series taking values from a finite or countably infinite set X. Assume that the distribution of the process is otherwise unknown. We propose a sequence of stopping times λ n along which we will be able to estimate the conditional probability P(Xλ n+1=x|X 0,...,λ n) from data segment (X 0,...,λ 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→∞ n/λ n > almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then λ n is upperbounded 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. Assume that the distribution of the process is otherwise unknown. We propose a sequence of stopping times λ n along which we will be able to estimate the conditional probability P(Xλ n+1=x|X 0,...,λ n) from data segment (X 0,...,λ 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→∞ n/λ n > almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then λ n is upperbounded by a polynomial, eventually almost surely.

KW - Nonparametric estimation

KW - Stationary processes

UR - http://www.scopus.com/inward/record.url?scp=17444391313&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=17444391313&partnerID=8YFLogxK

U2 - 10.1007/s00440-004-0386-3

DO - 10.1007/s00440-004-0386-3

M3 - Article

AN - SCOPUS:17444391313

VL - 132

SP - 1

EP - 12

JO - Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete

JF - Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete

SN - 0178-8051

IS - 1

ER -