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Doubly stochastic Hilbertian processes

Part of: Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Serge Guillas
Serge Guillas*
Affiliation:
Université Pierre et Marie Curie–Paris VI and École des Mines de Douai
*
∗ Current address: The University of Chicago, Center for Integrating Statistical and Environmental Science, Chicago, IL 60637, USA. Email address: [email protected]
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Abstract

In this paper, we consider a Hilbert-space-valued autoregressive stochastic sequence (Xn) with several regimes. We suppose that the underlying process (In) which drives the evolution of (Xn) is stationary. Under some dependence assumptions on (In), we prove the existence of a unique stationary solution, and with a symmetric compact autocorrelation operator, we can state a law of large numbers with rates and the consistency of the covariance estimator. An overall hypothesis states that the regimes where the autocorrelation operator's norm is greater than 1 should be rarely visited.

Keywords

Hilbertnonlinearautoregressivestability

MSC classification

Primary: 60G10: Stationary processes
Secondary: 62M10: Time series, auto-correlation, regression, etc.
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 3 , September 2002 , pp. 566 - 580
Copyright
Copyright © Applied Probability Trust 2002 

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References

[1]. Bass, J. (1955). Sur la compatibilité des fonctions de répartition. C. R. Acad. Sci. Paris 240, 839–841.Google Scholar
[2]. Besse, P., and Cardot, H. (1996). Approximation spline de la prévision d'un processus fonctionnel autorégressif d'ordre 1. Canad. J. Statist. 24, 467–487.Google Scholar
[3]. Besse, P., Cardot, H., and Stephenson, D. (2000). Autoregressive forecasting of some functional climatic variations. Scand. J. Statist. 27, 673–687.Google Scholar
[4]. Bosq, D. (1991). Modelization, nonparametric estimation and prediction for continuous time processes. In Nonparametric Functional Estimation and Related Topics (NATO ASI Ser. C 335), ed. Roussas, G., Kluwer, Dordrecht, pp. 509–529.Google Scholar
[5]. Bosq, D. (2000). Linear Processes in Function Spaces (Lecture Notes Statist. 149). Springer, New York.Google Scholar
[6]. Bosq, D. (2002). Estimation of mean and covariance operator of autoregressive processes in Banach spaces. Statist. Inf. Stoch. Process. 7, 1–20.Google Scholar
[7]. Bougerol, P., and Picard, N. (1992). Strict stationarity of generalized autoregressive processes. Ann. Prob. 20, 1714–1730.Google Scholar
[8]. Brandt, A. (1986). The stochastic equation Y n+1=A n Y n +B n with stationary coefficients. Adv. Appl. Prob. 18, 211–220.Google Scholar
[9]. Cavallini, A. et al. (1994). Nonparametric prediction of harmonic levels in electrical networks. In Proc. IEEE ICHPS 6, Bologna.Google Scholar
[10]. Damon, J., and Guillas, S. (2002). The inclusion of exogenous variables in functional autoregressive ozone forecasting. To appear in Environmetrics.Google Scholar
[11]. De Haan, L., Resnick, S., Rootzen, H., and de Vries, C. (1989). Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes. Stoch. Proces. Appl. 32, 213–224.Google Scholar
[12]. Francq, C. and Zakoïan, J.-M. (2001), Stationarity of multivariate Markov-switching ARMA models. J. Econometrics 102, 339–364 Google Scholar
[13]. Guillas, S. (2000). Non-causalité et discrétisation fonctionnelle, théorèmes limites pour un processus ARHX(1). C. R. Acad. Sci. Paris Math. 331, 91–94.Google Scholar
[14]. Horst, U. (2001). The stochastic equation Y t+1=AtYt+Bt with non-stationary coefficients. J. Appl. Prob. 38, 80–94.Google Scholar
[15]. Mélard, G., and Roy, R. (1988). Modèles de séries chronologiques avec seuils. Rev. Statist. Appl. 36, 5–24.Google Scholar
[16]. Nicholls, D. F., and Quinn, B. G. (1982). Random Coefficent Autoregressive Models: An Introduction (Lecture Notes Statist. 11). Springer, New York.Google Scholar
[17]. Petrucelli, J. D., and Woolford, S. W. (1984). A threshold AR(1) model. J. Appl. Prob. 21, 270–286.Google Scholar
[18]. Pourahmadi, M. (1986). On stationarity of the solution of a doubly stochastic model. J. Time Ser. Anal. 7, 123–131.Google Scholar
[19]. Rio, E. (2000). Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants (Math. Appl. 31). Springer, New York.Google Scholar
[20]. Tjösteim, D. (1986). Some doubly stochastic time series models. J. Time Ser. Anal. 7, 51–72.Google Scholar
[21]. Tjösteim, D. (1994). Non-linear time series: a selective review. Scand. J. Statist. 21, 97–130.Google Scholar
[22]. Tong, H. (1983). Threshold Models in Nonlinear Time Series Analysis (Lecture Notes Statist. 21). Springer, Berlin.Google Scholar
[23]. Tong, H. (1990). Nonlinear Times Series. Oxford University Press.Google Scholar
[24]. Yao, J. (2001). On square-integrability of an AR process with Markov switching. Statist. Prob. Lett. 52, 265–270.Google Scholar
[25]. Yao, J.-F., and Attali, J.-G. (2000). On stability of nonlinear AR processes with Markov switching. Adv. Appl. Prob. 32, 394–407.Google Scholar