Théorie géométrique de la Représentation Markovienne
Annales de l'institut Henri Poincaré. Section B. Calcul des probabilités et statistiques, Volume 16 (1980) no. 3, pp. 225-297.
@article{AIHPB_1980__16_3_225_0,
     author = {Ruckebusch, Guy},
     title = {Th\'eorie g\'eom\'etrique de la {Repr\'esentation} {Markovienne}},
     journal = {Annales de l'institut Henri Poincar\'e. Section B. Calcul des probabilit\'es et statistiques},
     pages = {225--297},
     publisher = {Gauthier-Villars},
     volume = {16},
     number = {3},
     year = {1980},
     zbl = {0441.60072},
     mrnumber = {595201},
     language = {fr},
     url = {http://www.numdam.org/item/AIHPB_1980__16_3_225_0/}
}
TY  - JOUR
AU  - Ruckebusch, Guy
TI  - Théorie géométrique de la Représentation Markovienne
JO  - Annales de l'institut Henri Poincaré. Section B. Calcul des probabilités et statistiques
PY  - 1980
DA  - 1980///
SP  - 225
EP  - 297
VL  - 16
IS  - 3
PB  - Gauthier-Villars
UR  - http://www.numdam.org/item/AIHPB_1980__16_3_225_0/
UR  - https://zbmath.org/?q=an%3A0441.60072
UR  - https://www.ams.org/mathscinet-getitem?mr=595201
LA  - fr
ID  - AIHPB_1980__16_3_225_0
ER  - 
%0 Journal Article
%A Ruckebusch, Guy
%T Théorie géométrique de la Représentation Markovienne
%J Annales de l'institut Henri Poincaré. Section B. Calcul des probabilités et statistiques
%D 1980
%P 225-297
%V 16
%N 3
%I Gauthier-Villars
%G fr
%F AIHPB_1980__16_3_225_0
Ruckebusch, Guy. Théorie géométrique de la Représentation Markovienne. Annales de l'institut Henri Poincaré. Section B. Calcul des probabilités et statistiques, Volume 16 (1980) no. 3, pp. 225-297. http://www.numdam.org/item/AIHPB_1980__16_3_225_0/

[Aka-1] H. Akaike, Stochastic theory of minimal realizations. I. E. E. E. Trans. Automatic Control, AC-19, 1974, p. 667-674. | MR | Zbl

[Aka-2] H. Akaike, Markovian representation of stochastic processes by canonical variables. S. I. A. M. J. Control., t. 13, 1975, p. 162-173. | MR | Zbl

[And] B.D.O. Anderson, The inverse problem of stationary covariance generation J. Statistical Physics, t. 1, 1969, p. 133-147. | MR

[AnK] B.D.O. Anderson et T. Kailath, Forwards, backwards and dynamically reversible Markovian models of second order processes. Proc. 1978 I. E. E. E. International Symposium on Circuits and Systems, New York, p. 981-986. | MR

[AnV] B.D.O. Anderson et S. Vongpanitlerd, Network analysis and synthesis. A modern System Theory approach, Prentice-Hall, New Jersey, 1973.

[BaB] J.S. Baras et R.W. Brockett, H2-functions and infinite dimensional realization theory. S. I. A. M. J. Control, t. 13, 1975, p. 221-241. | MR | Zbl

[BaD] J.S. Baras et P. Dewilde, Invariant subspace methods in linear multivariable-distributed systems and lumped-distributed network synthesis. Proc. I. E. E. E., t. 64, 1976, p. 160-178. | MR

[BBF] J.S. Baras, R.W. Brockett et P.A. Fuhrmann, State-space model for infinite dimensional systems. I. E. E.E. Trans. Automatic Control, AC-19,1974, p.693-700. | MR | Zbl

[Ben] A. Bensoussan, Filtrage optimal des systèmes linéaires, Dunod, Paris, 1971. | Zbl

[Bir] G. Birkhoff, Lattice theory. Amer. Math. Soc., Coll. Publ., vol. 25, Providence, Rhode Island, 1948. | MR | Zbl

[B1G] R.M. Blumenthal et R.K. Getoor, Markov processes and potential theory, Academic Press, New York, 1968. | MR | Zbl

[BoJ] G.E.P. Box et G.M. Jenkins, Time series analysis: forecasting and control, Holden Day, San Francisco, 1976. | MR | Zbl

[Bro] R.W. Brockett, Finite dimensional linear systems, J. Wiley and Sons, New York, 1970. | Zbl

[Cas] J.L. Casti, Dynamical systems and their applications: Linear theory. Mathematics in Science and Engineering, vol. 135, Academic Press, New York, 1977. | MR | Zbl

[Che] C.T. Chen, Introduction to linear system theory. H. R. W. Series in Electrical Engineering, Electronics and Systems, Holt, Rinehart and Winston, 1970.

[CuP] R.F. Curtain et A.J. Pritchard, Infinite dimensional linear systems theory. Lecture notes in Control and Information Science, vol. 8, Springer-Verlag, Berlin, 1978. | MR | Zbl

[Dav] M.C. Davis, Factoring the spectral matrix. I. E. E. E. Trans. Automatic Control, AC-8, 1963, p. 296-305.

[Doo-1] J.L. Doob, The elementary Gaussian processes. Annals of Math. Stat., t. 15, 1944, p. 229-282. | MR | Zbl

[Doo-2] J.L. Doob, Stochastic processes, J. Wiley and Sons, New York, 1953. | MR | Zbl

[DSS] R.G. Douglas, H.S. Shapiro et A.L. Shields, Cyclic vectors and invariant subspaces for the backward shift operator. Ann. Inst. Fourier (Grenoble), t. 20, 1970, p. 37-76. | Numdam | MR | Zbl

[Dun] T.E. Duncan, A geometric approach to linear control and estimation (à paraître).

[DuS] N. Dunford et J.T. Schwartz, Linear operators, vol. 1, Interscience, New York, 1957. | Zbl

[DyM] H. Dym et H.P. Mckean, Gaussian processes, function theory and the inverse spectral problem, Academic Press, New York, 1976. | MR | Zbl

[Fau-1] P. Faurre, Representation of stochastic processes, Ph. D. Dissertation, Stanford University, February 1967.

[Fau-2] P. Faurre, Réalisations Markoviennes de processus stationnaires. Rapport de recherche, n° 13, 1973, I. R. I. A., 78150 Le Chesnay.

[FCG] P. Faurre, M. Clerget et F. Germain, Opérateurs rationnels positifs. Application à l'hyperstabilité et aux processus aléatoires. Méthodes Mathématiques de l'informatique, t. 8, Dunod, Paris, 1979. | MR | Zbl

[Fuh-1] P.A. Fuhrmann, Realization theory in Hilbert space for a class of transfer functions. J. Funct. Anal., t. 18, 1975, p. 338-349. | MR | Zbl

[Fuh-2] P.A. Fuhrmann, Exact controllability and observability and realization theory in Hilbert space. J. of Math. Anal. and applications, t. 53, 1976, p. 377-392. | MR | Zbl

[Fuh-3] P.A. Fuhrmann, Linear operators and systems in Hilbert space (à paraître), McGraw-Hill. | Zbl

[Ger] F. Germain, Algorithmes de calcul de réalisations Markoviennes. Cas singuliers et stabilité. Rapport de recherche, n° 66, 1973, I. R. I. A., 78150 Le Chesnay.

[Hal] P.R. Halmos, Shifts on Hilbert spaces. J. Reine Angew. Math., t. 208, 1961, p. 102-112. | MR | Zbl

[Han] E.J. Hannan, Multiple time series, Wiley and Sons, New York, 1970. | MR | Zbl

[Hel-1] H. Helson, Lectures on invariant subspaces, Academic Press, New York, 1964. | MR | Zbl

[Hel-2] J.W. Helton, Discrete time systems, operator models and scattering theory. J. Funct. Anal., t. 16, 1974, p. 15-38. | MR | Zbl

[Hel-3] J.W. Helton, Systems with infinite-dimensional state space : the Hilbert space approach. Proc. I. E. E. E., t. 64, 1976, p. 145-160. | MR

[Hof] K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, Englewood Cliffs, New Jersey, 1962. | MR | Zbl

[IbR] I. Ibrahimov et Y.A. Rozanov, Processus aléatoires Gaussiens, Éditions de Moscou, 1974. | Zbl

[Kai] T. Kailath, A view of three decades of linear filtering theory. I. E. E. E. Trans. Info. Theory, IT-20, vol. 2, 1974, p. 146-181. | MR | Zbl

[KaG] T. Kailath et R.A. Geesey, An innovations approach to least-squares estimation. Part IV: Recursive estimation given lumped covariance functions. I. E. E. E. Trans. Automatic Control, AC-16, 1971, p. 720-727. | MR

[KMS] T. Kailath, M. Morf et S. Sidhu, Some new algorithms for recursive estimation in constant discrete-time linear systems. Proc. seventh Princeton Symp. on Information and system science, 1973, p. 344-352.

[Kal-1] R.E. Kalman, A new approach to linear filtering and prediction problems. Trans. Amer. Soc. Mech. Eng., J. Basic Engineering, t. 82, 1960, p. 35-45.

[Kal-2] R.E. Kalman, Linear stochastic filtering. Reappraisal and outlook. Proc. Symposium on System Theory, Polytechnic Institute of Brooklyn, 1965, p. 197-205. | MR

[Kal-3] R.E. Kalman, Lectures on controllability and observability. C. I. M. E. Summer course 1968, Cremonese, Roma, 1969, p. 1-149. | MR | Zbl

[KFA] R.E. Kalman, P.L. Falb et M.A. Arbib, Topics in Mathematical System Theory, McGraw-Hill, New York, 1969. | MR | Zbl

[Kat] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1966. | MR | Zbl

[Kol] A. Kolmogorov, Stationary sequences in Hilbert space (en russe). Bull. Math. Univ. Moscou, vol. 2, n° 6, 1941, p. 1-40. | MR | Zbl

[LaP] P.D. Lax et R.S. Phillips, Scattering theory, Academic Press, New York, 1967. | MR | Zbl

[Lin] A. Lindquist, A new algorithm for optimal filtering of discrete-time stationary processes. S. I. A. M. J. Control, t. 12, 1974, p. 736-746. | MR | Zbl

[LiP-1] A. Lindquist et G. Picci, On the stochastic realization problem. S. I. A. M. J. Control (à paraître). | Zbl

[LiP-2] A. Lindquist et G. Picci, On the structure of minimal splitting subspaces in stochastic realization theory. Proc. 1977 Conf. Decision and Control,New Orleans, 1977, p. 42-48. | MR

[LiP-3] A. Lindquist et G. Picci, A state-space theory for stationary stochastic processes. Proc. 21st Midwest Symposium on Circuits and Systems, Ames, Iowa, August 1978.

[LiP-4] A. Lindquist et G. Picci, A Hardy space approach to the stochastic realization problem. Proc. 1978 Conf. Decision and Control, San Diego, January 1979. | MR | Zbl

[LiP-5] A. Lindquist et G. Picci,Realization theory for multivariate stationary Gaussian processes I: State space construction. 4th Int. Symposium on the Mathematical theory of Networks and Systems, Delft (Pays-Bas), 1979, p. 140-148. | Zbl

[LPR] A. Lindquist, G. Picci et G. Ruckebusch, On minimal splitting subspaces and Markovian representations. Mathematical Systems Theory, t. 12, 1979, p. 271-279. | MR | Zbl

[LjK] L. Ljung et T. Kailath, Backwards Markovian models for second-order stochastic processes. I. E. E. E. Trans. Info. Theory, IT-22, 1976, p. 488-491. | MR | Zbl

[Man] V. Mandrekar, On multivariate wide-sense Markovprocesses. Nagoya Math. J., vol. 33, 1968, p. 7-19. | MR | Zbl

[Mas] P. Masani, On helixes in Hilbert space I. Theory of Proba. and its applications, vol. 17, 1972, p. 1-19. | MR | Zbl

[MaR] P. Masani et J. Robertson, The time-domain of continuous parameter weakly stationary stochastic processes. Pacific J. Math., t. 12, 1962, p. 1361-1378. | MR | Zbl

[MeL] R.K. Mehra et D.G. Lainiotis, Systems identification: advances and case studies, Academic Press, New York, 1977.

[MeP] M. Metivier et J. Pellaumail, Stochastic Integration, Academic Press, à paraître. | Zbl

[Mey] P.A. Meyer, Processus de Markov, Springer-Verlag, Berlin, 1967. | MR | Zbl

[Nev-1] J. Neveu, Processus aléatoires Gaussiens, Presses de l'Université de Montréal, 1968. | MR | Zbl

[Nev-2] J. Neveu, Intégrales stochastiques et applications, Cours de 3e cycle, Université de Paris VI, 1971-1972.

[Pic] G. Picci, Stochastic realization of Gaussian processes. Proc. I. E. E. E., t. 64, 1976, p. 112-122. | MR

[Pot] V.P. Potapov, The multiplicative structure of J contractive matrix functions. Amer. Math. Soc. Transl., t. 15, 1960, p. 131-243. | MR | Zbl

[RiS] F. Riesz et B. Sz.-Nagy, Leçons d'analyse fonctionnelle, Gauthier-Villars, Paris, 1965. | Zbl

[Roz-1] Y.A. Rozanov, Stationary random processes, Holden Day, San Francisco, 1967. | MR | Zbl

[Roz-2] Y.A. Rozanov, On two selected topics connected with Stochastic System Theory. Applied Math. and Optimization, vol. 3, n° 1, 1976, p. 73-80. | MR | Zbl

[Roz-3] Y.A. Rozanov, On Markov extensions of a random process. Theory of Proba. and its applications, vol. 22, 1977, p. 190-195. | Zbl

[Ruc-1] G. Ruckebusch, Représentations Markoviennes de processus Gaussiens stationnaires. Thèse de 3e cycle, Lab. de Calcul des Probabilités, Université de Paris VI, mai 1975.

[Ruc-2] G. Ruckebusch, Représentations Markoviennes de processus Gaussiens stationnaires. C. R. Acad. Sciences, Paris, Série A, t. 282, 1976, p. 649-651. | MR | Zbl

[Ruc-3] G. Ruckebusch, Représentations Markoviennes de processus Gaussiens stationnaires et applications statistiques. Journées de Statistique des processus stochastiques, Grenoble, 1977, Springer-Verlag, Berlin, Lecture Notes in Mathematics, t. 636, p. 115-139. | Zbl

[Ruc-4] G. Ruckebusch, On the theory of Markovian Representation. Proc. Measure theory-applications to stochastic analysis, Oberwolfach (Allemagne), 1977, Springer-Verlag, Berlin, Lecture Notes in Mathematics, t. 695, p. 77-87. | MR | Zbl

[Ruc-5] G. Ruckebusch, A state space approach to the stochastic realization problem. Proc. 1978 I. E. E. E. International Symposium on Circuits and Systems, New York, p. 972-977.

[Ruc-6] G. Ruckebusch, Factorisations minimales de densités spectrales et représentations Markoviennes. Proc. 1er Colloque A. F. C. E. T.-S. M. F. de Mathématiques Appliquées, t. I, École Polytechnique, 91128 Palaiseau, 1978, p. 365-375. | Zbl

[Ruc-7] G. Ruckebusch, A geometric approach to the stochastic realization problem. Rapport interne, n° 41, Centre de Math. Appliquées, École Polytechnique, 91128 Palaiseau, novembre 1978, soumis à publication.

[Ruc-8] G. Ruckebusch, Observability and constructibility in the theory of Markovian representation. 4th Int. Symposium on the Mathematical theory of Networks and Systems, Delft (Pays-Bas), 1979, p. 128-132. | Zbl

[Ruc-9) G. Ruckebusch, Théorie géométrique de la Représentation Markovienne. Thèse d'État, Lab. de Calcul des Probabilités, Université de Paris VI, avril 1980.

[Sal] G. Salut, Identification optimale des systèmes linéaires stochastiques. Thèse d'État, Université Paul-Sabatier, Toulouse, 1976.

[SzF-1] B. Sz.-Nagy et C. Foias, Opérateurs sans multiplicité. Acta Sci. Math., t. 30, 1969, p. 1-18. | MR | Zbl

[SzF-2] B. Sz.-Nagy et C. Foias, Harmonic analysis of operators on Hilbert space, North Holland, Amsterdam, 1970. | MR | Zbl

[Tay] A.E. Taylor, Introduction to functional analysis, John Wiley and Sons, New York, 1958. | MR | Zbl

[Urb] K. Urbanik, Lectures on Prediction Theory, Springer-Verlag, New York, 1967. | MR | Zbl

[VaV] C. Van Putten et J.H. Van Schuppen, On stochastic dynamical systems. 4th Int. Symposium on the Mathematical Theory of Networks and Systems, Delft (Pays-Bas), 1979, p. 350-356. | Zbl

[WiM] N. Wiener et P. Masani, The prediction theory of multivariate stochastic processes, Part I. Acta Math., t. 98, 1957, p. 111-150; Part II, Acta Math., t. 99, 1958, p. 93-137. | Zbl

[Wil-1] J.C. Willems, Least squares stationary optimal control and the algebraic Riccati equation. I. E. E. E. Trans. Automatic Control, AC-16, 1971, p. 621-634. | MR

[Wil-2] J.C. Willems, Dissipative dynamical systems, Part II: Linear systems with quadratic supply rates. Archive for rational Mechanics and Analysis, t. 45, 1972, p. 352-393. | MR | Zbl

[WiV] J.C. Willems et J.H. Van Schuppen, Stochastic systems and the problem of state space realization (à paraître).

[Wol] H. Wold, A study in the analysis of stationary time series, Almquist and Wiksell, Stockholm, 1938. | JFM | MR | Zbl

[Won] W.M. Wonham, Linear multivariable control. A geometric approach, Springer-Verlag, Berlin, Lecture Notes in Economics and Mathematical Systems, t. 101, 1974. | MR | Zbl

[Yos] K. Yosida, Functional analysis, Springer-Verlag, Berlin, 1974. | MR | Zbl