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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.

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