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Modelization, nonparametric estimation and prediction for continuous time processes. (English) Zbl 0737.62032
Nonparametric functional estimation and related topics, NATO ASI Ser., Ser. C 335, 509-529 (1991).
[For the entire collection see Zbl 0722.00032.]
From the author’s abstract: In order to predict a continuous-time process on an entire time-interval, a Hilbert-space-valued autoregressive process $$X_ t$$, $$t=0,\pm1,\pm2,\dots$$ is studied, presenting consistent nonparametric estimators for the covariance operator $$C$$ of $$X_ t$$ and the cross estimator covariance of $$(X_ t,X_{t+1})$$. Then an estimator $$\rho_ n$$ of the autocorrelation operator is constructed by projecting the data on a finite dimensional subspace generated by the eigenvectors of $$C$$. Under mild regularity conditions it is shown that the predictor based upon $$\rho_ n$$ converges in probability. Assuming an additional mixing condition for $$(X_ t)$$, almost sure convergence is obtained.

MSC:
 62G07 Density estimation 62M09 Non-Markovian processes: estimation