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
Prediction for discrete time series. / Morvai, G.; Weiss, Benjamin.
In: Probability Theory and Related Fields, Vol. 132, No. 1, 05.2005, p. 1-12.Research output: Contribution to journal › Article
}
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 -