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June, 1975 Prediction from a Random Time Point
Georg Lindgren
Ann. Probab. 3(3): 412-423 (June, 1975). DOI: 10.1214/aop/1176996349


In prediction (Wiener-, Kalman-) of a random normal process $\{X(t), t \in R\}$ it is normally required that the time $t_0$ from which prediction is made does not depend on the values of the process. If prediction is made only from time points at which the process takes a certain value $u,$ given a priori, ("prediction under panic"), the Wiener-prediction is not necessarily optimal; optimal should then mean best in the long run, for each single realization. The main theorem in this paper shows that when predicting only from upcrossing zeros $t_\nu$, the Wiener-prediction gives optimal prediction of $X(t_\nu + t)$ as $t_\nu$ runs through the set of zero upcrossings, if and only if the derivative $X'(t_\nu)$ at the crossing points is observed. The paper also gives the conditional distribution from which the optimal predictor can be computed.


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Georg Lindgren. "Prediction from a Random Time Point." Ann. Probab. 3 (3) 412 - 423, June, 1975.


Published: June, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0311.60029
MathSciNet: MR397858
Digital Object Identifier: 10.1214/aop/1176996349

Primary: 60G25
Secondary: 60G40 , 62M20

Keywords: prediction , stopping times , zero-crossings

Rights: Copyright © 1975 Institute of Mathematical Statistics

Vol.3 • No. 3 • June, 1975
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