Théorie géométrique de la Représentation Markovienne
Annales de l'I.H.P. Probabilités et statistiques, Volume 16 (1980) no. 3, p. 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'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {16},
     number = {3},
     year = {1980},
     pages = {225-297},
     zbl = {0441.60072},
     mrnumber = {595201},
     language = {fr},
     url = {http://www.numdam.org/item/AIHPB_1980__16_3_225_0}
}
Ruckebusch, Guy. Théorie géométrique de la Représentation Markovienne. Annales de l'I.H.P. 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 426914 | Zbl 0292.93013

[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 449849 | Zbl 0312.60017

[And] B.D.O. Anderson, The inverse problem of stationary covariance generation J. Statistical Physics, t. 1, 1969, p. 133-147. | MR 253768

[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 522111

[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 456729 | Zbl 0297.93010

[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 416679

[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 434520 | Zbl 0291.93006

[Ben] A. Bensoussan, Filtrage optimal des systèmes linéaires, Dunod, Paris, 1971. | Zbl 0231.93022

[Bir] G. Birkhoff, Lattice theory. Amer. Math. Soc., Coll. Publ., vol. 25, Providence, Rhode Island, 1948. | MR 29876 | Zbl 0033.10103

[B1G] R.M. Blumenthal et R.K. Getoor, Markov processes and potential theory, Academic Press, New York, 1968. | MR 264757 | Zbl 0169.49204

[BoJ] G.E.P. Box et G.M. Jenkins, Time series analysis: forecasting and control, Holden Day, San Francisco, 1976. | MR 436499 | Zbl 0363.62069

[Bro] R.W. Brockett, Finite dimensional linear systems, J. Wiley and Sons, New York, 1970. | Zbl 0216.27401

[Cas] J.L. Casti, Dynamical systems and their applications: Linear theory. Mathematics in Science and Engineering, vol. 135, Academic Press, New York, 1977. | MR 510991 | Zbl 0555.93002

[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 516812 | Zbl 0389.93001

[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 10931 | Zbl 0060.28907

[Doo-2] J.L. Doob, Stochastic processes, J. Wiley and Sons, New York, 1953. | MR 58896 | Zbl 0053.26802

[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 270196 | Zbl 0186.45302

[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 0084.10402

[DyM] H. Dym et H.P. Mckean, Gaussian processes, function theory and the inverse spectral problem, Academic Press, New York, 1976. | MR 448523 | Zbl 0327.60029

[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 525380 | Zbl 0424.93002

[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 367768 | Zbl 0304.93008

[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 504097 | Zbl 0342.93009

[Fuh-3] P.A. Fuhrmann, Linear operators and systems in Hilbert space (à paraître), McGraw-Hill. | Zbl 0456.47001

[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 152896 | Zbl 0107.09802

[Han] E.J. Hannan, Multiple time series, Wiley and Sons, New York, 1970. | MR 279952 | Zbl 0211.49804

[Hel-1] H. Helson, Lectures on invariant subspaces, Academic Press, New York, 1964. | MR 171178 | Zbl 0119.11303

[Hel-2] J.W. Helton, Discrete time systems, operator models and scattering theory. J. Funct. Anal., t. 16, 1974, p. 15-38. | MR 445310 | Zbl 0282.93033

[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 416694

[Hof] K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, Englewood Cliffs, New Jersey, 1962. | MR 133008 | Zbl 0117.34001

[IbR] I. Ibrahimov et Y.A. Rozanov, Processus aléatoires Gaussiens, Éditions de Moscou, 1974. | Zbl 0291.60021

[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 465437 | Zbl 0307.93040

[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 309259

[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 256769

[Kal-3] R.E. Kalman, Lectures on controllability and observability. C. I. M. E. Summer course 1968, Cremonese, Roma, 1969, p. 1-149. | MR 280225 | Zbl 0208.17201

[KFA] R.E. Kalman, P.L. Falb et M.A. Arbib, Topics in Mathematical System Theory, McGraw-Hill, New York, 1969. | MR 255260 | Zbl 0231.49001

[Kat] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1966. | MR 203473 | Zbl 0148.12601

[Kol] A. Kolmogorov, Stationary sequences in Hilbert space (en russe). Bull. Math. Univ. Moscou, vol. 2, n° 6, 1941, p. 1-40. | MR 9098 | Zbl 0063.03291

[LaP] P.D. Lax et R.S. Phillips, Scattering theory, Academic Press, New York, 1967. | MR 217440 | Zbl 0186.16301

[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 408968 | Zbl 0296.93037

[LiP-1] A. Lindquist et G. Picci, On the stochastic realization problem. S. I. A. M. J. Control (à paraître). | Zbl 0428.60052

[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 504278

[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 528901 | Zbl 0428.60049

[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 0496.93021

[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 529562 | Zbl 0401.60056

[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 424350 | Zbl 0334.93037

[Man] V. Mandrekar, On multivariate wide-sense Markovprocesses. Nagoya Math. J., vol. 33, 1968, p. 7-19. | MR 248909 | Zbl 0174.49602

[Mas] P. Masani, On helixes in Hilbert space I. Theory of Proba. and its applications, vol. 17, 1972, p. 1-19. | MR 298468 | Zbl 0283.60032

[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 149562 | Zbl 0113.33403

[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 0463.60004

[Mey] P.A. Meyer, Processus de Markov, Springer-Verlag, Berlin, 1967. | MR 219136 | Zbl 0189.51403

[Nev-1] J. Neveu, Processus aléatoires Gaussiens, Presses de l'Université de Montréal, 1968. | MR 272042 | Zbl 0192.54701

[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 429288

[Pot] V.P. Potapov, The multiplicative structure of J contractive matrix functions. Amer. Math. Soc. Transl., t. 15, 1960, p. 131-243. | MR 114915 | Zbl 0090.05403

[RiS] F. Riesz et B. Sz.-Nagy, Leçons d'analyse fonctionnelle, Gauthier-Villars, Paris, 1965. | Zbl 0122.11205

[Roz-1] Y.A. Rozanov, Stationary random processes, Holden Day, San Francisco, 1967. | MR 214134 | Zbl 0152.16302

[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 444260 | Zbl 0355.93010

[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 0379.60038

[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 400372 | Zbl 0367.60038

[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 0387.60043

[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 527075 | Zbl 0409.93038

[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 0497.93033

[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 0499.93013

[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 420316 | Zbl 0174.18301

[SzF-2] B. Sz.-Nagy et C. Foias, Harmonic analysis of operators on Hilbert space, North Holland, Amsterdam, 1970. | MR 275190 | Zbl 0201.45003

[Tay] A.E. Taylor, Introduction to functional analysis, John Wiley and Sons, New York, 1958. | MR 98966 | Zbl 0081.10202

[Urb] K. Urbanik, Lectures on Prediction Theory, Springer-Verlag, New York, 1967. | MR 224158 | Zbl 0262.60023

[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 0504.93054

[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 0080.13002

[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 308890

[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 527463 | Zbl 0252.93003

[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 64.1200.02 | MR 61344 | Zbl 0019.35602

[Won] W.M. Wonham, Linear multivariable control. A geometric approach, Springer-Verlag, Berlin, Lecture Notes in Economics and Mathematical Systems, t. 101, 1974. | MR 378912 | Zbl 0314.93007

[Yos] K. Yosida, Functional analysis, Springer-Verlag, Berlin, 1974. | MR 350358 | Zbl 0286.46002