Critères de mélange pour des processus stationnaires : estimation sous des hypothèses de mélange : entropie des processus linéaires

Mokkadem, Abdelkader

Directeur de thèse : Dacunha-Castelle, Didier

Membres du jury : Dacunha-Castelle, Didier ; Bretagnolle, Jean ; Elie, Laure ; Guyon, Xavier ; Le Breton, Alain ; Rais, Mustapha

Thèses d'Orsay, no. 213 (1987) , 230 p.

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@phdthesis{BJHTUP11_1987__0213__P0_0,
     author = {Mokkadem, Abdelkader},
     title = {Crit\`eres de m\'elange pour des processus stationnaires : estimation sous des hypoth\`eses de m\'elange : entropie des processus lin\'eaires},
     series = {Th\`eses d'Orsay},
     publisher = {Universit\'e Paris-Sud Centre d'Orsay},
     number = {213},
     year = {1987},
     language = {fr},
     url = {http://archive.numdam.org/item/BJHTUP11_1987__0213__P0_0/}
}

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Mokkadem, Abdelkader. Critères de mélange pour des processus stationnaires : estimation sous des hypothèses de mélange : entropie des processus linéaires. Thèses d'Orsay, no. 213 (1987), 230 p. http://numdam.org/item/BJHTUP11_1987__0213__P0_0/

Bibliographie
Cité par

[1] A. Mokkadem - Sur un modèle autorégressif non linéaire, ergodicité et ergodicité géométrique. J.T.S.A., vol.8, n°2, 1987, 195-204. | MR | Zbl

[2] A. Mokkadem - Le modèle non Linéaire AR(1) général. Ergodicité et ergodicité géométrique. C.R.A.S., t.301, série I, 1985, 889-892. | MR | Zbl

[3] A. Mokkadem - Conditions suffisantes d'existence et d'engodicité géométrique des modèles bilinéaires. C.R.A.S., t.301, sèrie I, 1985, 375-377. | MR | Zbl

[4] A. Mokkadem - Mixing properties of ARMA processes. submitted to Stoch. Proc. Appl. | MR | Zbl | DOI

[5] A. Mokkadem - Sur le mélange d'un processus ARMA vectoriel. C.R.A.S., t.303, série I, 1986, 519-521. | MR | Zbl

[6] A. Mokkadem - Propriétés de mélange des processus autorégressifs polynomiaux. Prépublication. Soumis aux Annales de l'I.H.P. | MR | Zbl | Numdam

[7] A. Mokkadem - Study of risks of kernel estimators. To appear | Zbl | MR | DOI

[8] A. Mokkadem - Etude des risques du estimateurs à noyaux. C.R.A.S., t.301, série I, 1985, 447-450. | MR | Zbl

[9] A. Mokkadem - Estimation of the entropy and information of absolutely continuous random variables. To appear in I.E.E.E. Trans. Inf. Theory. | MR | Zbl | DOI

[10] A. Mokkadem - Entropie des processus linéaires. Prépublication. Soumis à Prob. and Math. Stat. | MR | Zbl

[11] A. Mokkadem - Entropie de processus et erreur de prédiction. C.R.A.S., t.298, série I, 1984, 493-496. | MR | Zbl

[12] E. Nummelin, P. Tuominen - Geometric ergodicity of Harris-recurrent Markov chains. Stoch. Proc. Appl., 3, 1982, 187-202. | MR | Zbl

[13] R.L. Tweedie - Sufficient conditiom for ergodicity and recurrence of Markov chains. Stoch. Proc. Appl., 3, 1975, 385-403. | MR | Zbl

[14] Y.A. Davydov - Mixing conditions for Markov chains. Th. Prob. Appl., 28, 1973, 313-328. | MR | Zbl

[15] V.A. Volkonskii, Y.A. Rozanov - Some limit theorems for random functions, part II. Th. Prob. Appl., 6, 1961, 186-198. | Zbl

[16] M. Rosenblatt - A central limit theorem and a strong mixing condition. Proc. Nat. Acad.Sci.,U.S.A., 42, 1956, 43-47. | MR | Zbl | DOI

[17] I.A. Ibragimov, Y.V. Linnik - Independent and stationaly sequences of random variables. Walter-Noordhoof publishing, 1974. | MR | Zbl

[18] H. Takata - $L_{\infty}$ bounds for asymptotic normality of weakly dependent summands using Stein's methods. Ann. Prob., 9, 1981, 676-683. | MR | Zbl

[19] K. Yoshihara - Limiting behavior of $U$ -statistics for stationaly absolutely regular processes. Z. Wahr. Verw. Gebiete, 35, 1976, 237-252. | MR | Zbl | DOI

[20] R.C. Bradley - Absolute regularity and functions of Markov chains, Stoch. Proc. Appl., 14, 1983, 67-77. | MR | Zbl

[21] H.C.P. Berbée - Random walks with stationaly increments and renewal theory. Mathematical Center Tract n°112, AMSTERDAM, 1979. | MR | Zbl

[22] I.A. Ibragimov, Y. Rozanov - Processus aléatoires gaussiens. MIR, Moscou, 1974. | Zbl

[23] K.C. Chandra - Strong mixing properties of linear stochastic processes. J.A.P., 11, 1974, 401-408. | MR | Zbl

[24] V.V. Gorodetski - On the strong mixing property for linear sequences. Th. Prob. Appl., 22, 1977, 411-413. | Zbl

[25] C.S. Withers - Conditions for linear processes to be strong mixing. Z. Wahr. Verw. Gebiete, 1981, 477-480. | MR | Zbl | DOI

[26] T.D. Pham, L.T. Tran - Some mixing properties time series models. Stoch. Proc. Appl. 19, 1985, 297-303. | MR | Zbl

[27] J.L. Doob - Stochastic processes. Wiley, New-York, 1953. | MR | Zbl

[28] M. Rosenblatt - Markov processes. Structure and asymptotic behaviour. Springer, Berlin, 1971. | MR | Zbl

[29] H. Akaike - Markovian representation of stochastic processes. Ann. Inst. Stat. Math., 26, 1974, 363-387. | MR | Zbl

[30] M. Rosenblatt - Remarks on some non parametric estimates of a density function. Ann. Math. Stat., 27, 1956, 832-835. | MR | Zbl

[31] L. Devroye, L. Györfi - Non parametric density estimation. The $L_{1}$ view. Wiley, New-York, 1985. | MR | Zbl

[32] P. Doukhan, J. Leon, F. Portal - Principe d'invariance faible pour la mesure empirique d'une suite de variables aléatoires mélangeantes. Prépublication d'Orsay, 1985. | MR | Zbl

[33] I.A. Ahmad, P.E. Lin - A nonparametric estimation of the entropy for absolutely continuous distributions. I.E.E.E. Trans. Inf. Theory, vol. I, t.22, 1976, 372-375. | MR | Zbl

[34] B.L.S. Prakasa Rao - Non parametric functional, estimation. Academic Press, 1983. | MR | Zbl

[35] O. Vasicek - A test for normality based on sample entropy. J.R.S.S., serie B, 38, 1976, 54-59. | MR | Zbl

[36] S. Cambanis - Complex symetric stable variables and processes. In : Contribution to Statistics : Essays in Honor of Norman L. Johnson. ed. Sen, P.K., New-York, North-Holland, 1982, 63-79. | MR | Zbl

[37] C.E. Shannon - The mathematical theory of communications. Bell System Technical Journal, 1948. | MR | Zbl

[38] M. Kanter - Lower bound for non Linear prediction error in moving average processes. Ann. Prob., 7, 1979, 128-138. | MR | Zbl

[39] L.A. Shepp, D. Slepian, A.D. Wyner - On prediction of moving average Processes. Bell System Technical Journal, 59, 1980, 367-415. | MR | Zbl | DOI

[40] J.P. Burg - A new analysis technique for time series data. In modern spectrum analysis. ed. D.G. Childer, Wiley, New-York, 1978.

Doukhan P., Ghindes M. (1980 a), CRAS-Série I, T.290, 921-923. | Zbl

Doukhan P., Ghindes M. (1980 b), CRAS-Série I, T.291, 81-83.

Hall P., Heyde C.C. (1980), Martingale limit theory and its application. Academic Press. | MR | Zbl

Mokkadem A. (1985 a), CRAS-Série I, T. 301, 375-377.

Mokkadem A. (1985 b), CRAS-Série I, T. 301, 447-450. | Zbl

Nummelin E., Tuominen P. (1982), Geometric ergodicity of Harris recurrent Markov chains. Stoch. Proc. Appl. 12, 187-202. | MR | Zbl

Rosenblatt M. (1971), Markov Processes. Structure and asymptotic behaviour. Springer Verlag. | MR | Zbl

Tuan D. Pham, Lanh T. Tran (1980), CRAS-Série I, T. 290, 335-338. | Zbl

Tweedie R.L. (1975), Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space. Stoch. Proc. Appl. 3, 385-403. | MR | Zbl

Tweedie R.L. (1976), Criteria for classifying general Markov chains. Adv. Appl. Prob. 8, 737-771. | MR | Zbl

Tweedie R.L. (1983), The existence of moments for stationary Markov chains, J. Appl. Prob. 20, 191-196. | MR | Zbl | DOI

Yokoyama R. (1980), Moments bounds for stationary mixing sequences. Z. Wahr. verw. Gebiete 52, 45-57. | MR | Zbl | DOI

[1] G. Collomb et P. Doukhan. Comptes rendus. 296. série I. 1983. p 859-862. | Zbl

[2] P. Doukhan et M. Ghindes. Comptes rendus. 290, série A. 1980. p 921-923. | MR | Zbl

[3] A. Mokkadem. Prépublication, 1985

[4] A. Mokkadem. Comptes rendus. 301. série I. 1985. p. 447-450 | Zbl

[5] D. F. Nicholls et B. G. Qinn. Random coefficient autoregressive model: An introduction. Lectures Notes in Statisttes. 11. 1982. | MR | Zbl

[6] E. Nummelin et P. Tuominen Geometrie ergodicity of Harrris recurrent Markov chains. Stoch Proc. Appl. 12. 1982. P.187-202. | MR | Zbl

[7] S. Orey. Limit theorems for Markov chain transition probabilities. Van Nostrand. London 1971. | MR | Zbl

[8] D. Revuz. Markov chains. North Holland. Amsterdam. 1984. | MR | Zbl

[9] M. Rosenblatt. Markov processes. Structure and asymptotic Behaviour. Springer Verlag. 1971. | MR | Zbl

[10] D. Tuan Pham et Lanh T. Tran. Comptes rendus. 290. Series A. 1980. p. 335-338. | Zbl

[11] R. L. Tweedie. R-theory for Markov chains on general state space I. Ann. Prob., 2, 1974. p. 840-864. | MR | Zbl

(12] R. L. Tweedie, Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space. stoch. Proc. Appl., 3, 1975, p. 385-403. | MR | Zbl

[13] R. L. Tweedie, The existence of moments for stationary Markov chains. J. Appl. Prob., 20, 1983. p. 191-196. | MR | Zbl | DOI

[1] M. Bhaskara Rao, T. Subra Rao et A. M. Walker, On the existence of some bilinear time series models, J.T.S.A., 4, 1983, p. 95-110. | MR | Zbl

[2] S. Orey, Limit theorems for Markov chains, New York.

[3]Tuan Pham Dinh, Bilinear markovian representation and bilinear models, Technical report n° 161 UMIST, 1983. | MR | Zbl

[4] E. Nummelin et P. Tuominen, Geometric ergodicity of Harris recurrent Markov chains with applications to renewal theory, Stoch. Proc. Appl., 12, 1982, p. 187-202. | MR | Zbl

[5] R. L. Tweedie, Sufficient conditions for ergodicity and recurrence of Markov chains, Stoch. Proc. Appl., 3, 1975. p. 385-403. | MR | Zbl

[6] R. L. Tweedie, The existence of moments for stationary Markov chains, J. Appl. Prob. 20, 1983, p 191-196 | MR | Zbl | DOI

[1] Akaïke, H. - Markovian repersentation of stochastic processes. Ann. Inst. Stat. Math. 26 (1974) 363-387. | MR | Zbl

[2] Davydov, Yu, A. - Mixing conditions for Markov chains. Th. Prob. Appl. 18 (1973) 312-328. | Zbl | MR

[3] Gorodetskii, V.V. - On the strong mixing property for linear sequences. Th. Prob. Appl. 22 (1977) 411-413. | Zbl | MR

[4] Ibragimov, I.A., Rozanov, Y. - Processus aléatoires gaussiens. MIR. Moscou | Zbl

[5] Mokkadem, A. - Sur un modèle autorégressif non linéaire. Ergodicité et ergodicité géométrique. J.T.S.A., vol. 8, n° 2 (1987) 195-204. | MR | Zbl

[6] Nummelin, E., Tuominen, P. - Geometrie ergodicity of Harris-recurrent Markov chains. Stoch. Proc. Appl. 12 (1982) 187-202. | MR | Zbl

[7] Tuan D. Pham, Lanh T. Tran - Some mixing properties of time series models. Stoch. Proc. Appl. 19 (1986) 297-303. | MR | Zbl

[8] Tweedie, R.L. - Sufficient conditions for ergodicity and recurrence of Markov chains. Stoch. Proc. Appl. 3 (1975) 385-403. | MR | Zbl

[9] Tweedie, R.L. - Criteria for classifying general Markov chains. Adv. Appl. Prob. 8 (1976) 737-771. | MR | Zbl

[10] Withers, C.S. - Conditions for linear processes to be strong mixing. Z. Wahr. Verw. Gebiete 57(1981) 481-494. | MR | Zbl | DOI

[1] H. Akaike, Markovian representation of stochastic processes, Ann. Inst. Stat. Math., 26 (1974), 363-387. | MR | Zbl

[2] R. Azencott et D. Dacunha-Castelle, Séries d'observations irrégulières, Masson, Paris (1984). | MR | Zbl

[3] J.R. Blum, D.L. Hanson, L.H. Koopmans, On the strong law of large numbers for a class of stochastics processes, Z. Wahr. Verw. Gebiete, 2 (1963), 1-11. | MR | Zbl | DOI

[4] Th. Bröcker, Differentiable germs and catastrophes, Cambridge University Press (1975). | MR | Zbl | DOI

[5] Y.A. Davydov, Mixing conditions for Markow chains, Th. Prob. Appl., 28 (1973), 313-328. | Zbl | MR

[6] H. Delfs and M. Knebush, Semi algebraic topology over a real closed fields I et II, Math. Zeit., n° 177, 107-129 et n° 178, 175-213 (1981). | MR | Zbl

[7] J. Dieudonné, Eléments d'analyse tome III, Gauthier-Villars, Paris (1970). | MR | Zbl

[8] J.L. Doob, Stochastic processes, Wiley, New York (1953). | MR | Zbl

[9] V.V. Gorodetski, On the strong mixing property for linear sequences, Th. Prob. Appl., 22, 411-413 (1977). | Zbl

[10] P. Hall and C.C. Heyde, Martingale limit theory and its application, London Academic (1980). | MR | Zbl

[11] R.M. Hardt, Semi algebraic local triviality in semi algebraic mappings, Ann. Journ. Math., 102, 291-302. | MR | Zbl | DOI

[12] H. Hironaka, Resolution of singulariries of an algebraic variety, I-II, Ann. Math., 79, 109-326 (1964). | MR | Zbl

[13] I.A. Ibragimov and Y.V. Linnik, Independant and stationary sequences of random variables, Walth-Noordhoof publishing Gröningen (1974). | MR | Zbl

[14] I.A. Ibragimov et Y. Rozanov, Processus aléatoires gaussiens, MIR, Moscou (1974). | Zbl

[15] N. Jain and B. Jamison, Contributions to Doeblin's theory of Markov processes, Z. Wahr. verw. Gebiete, 8, 19-40 (1967). | MR | Zbl | DOI

[16] T.Y. Lam, An introduction to real algebra, Rocky Mountain Journ. Math., 14, 4 (1984). | MR | Zbl

[17] S. Lojasiewicz, Ensembles semi analytiques, multigraphie de l'I.H.E.S., Bures/Yvette (1965).

[18] A. Mokkadem, Sur le mélange d'un processus ARMA vectoriel, CRAS, série I, t. 303, 519-521 (1986). | MR | Zbl

[19] A. Mokkadem, Mixing properties of ARMA processes, soumis à Stoch.Proc.Appl. | MR | Zbl | DOI

[20] A. Mokkadem, Sur un modèle autorégressif non linéaire, ergodicité et ergodicité géométrique, J.T.S.A., 8, 195-204 (1987). | MR | Zbl

[21] A. Mokkadem, Conditions suffisantes d'existence et d'ergodicité géométrique des modèles bilinéaires, CRAS, série I, t. 301, 375-377 (1985). | MR | Zbl

[22] D. Mumford, Algebraic geometry I, Complex projective varieties, Springer Verlag, Berlin (1976). | MR | Zbl

[23] J. Nash, Real algebraic manifolds, Ann. Math., 56, 405-421 (1952). | MR | Zbl

[24] E. Nummelin and P. Tuominen, Geometric ergodicity of Harris recurrent Markov chains, Stoch. Proc. Appl., 12, 187-202 (1982). | MR | Zbl

[25] S. Orey, Limit theorems for Markov chain transition probabilities, Van Nostrand, London (1971). | MR | Zbl

[26] T.D. Pham and L.T. Tran, Some mixing properties of time series models, Stoch. Proc. Appl., 19, 297-303 (1986). | MR | Zbl

[27] T.D. Pham, Bilinear markovian representation and bilinear models, Stoch. Proc. Appl., 20, 295-306 (1985). | MR | Zbl

[28] M.Q. Pham, Introduction à la géométrie des variétés différentiables, Dunod, Paris (1969). | MR | Zbl

[29] D. Revuz, Markov chains, North Holland, Amsterdam (1984). | MR | Zbl

[30] M. Rosenblatt, Markov processes. Structure and asymptotic behaviour. Springer, Berlin (1971). | MR | Zbl

[31] M. Rosenblatt, A central limit theorem and a strong mixing condition, Proc. Nat. Acad. Sc. U.S.A., 42, 43-47 (1956). | MR | Zbl | DOI

[32] A. Seidenberg, A new decision method for elementary algebra, Ann. Math., 2, 365-374 (1952). | MR | Zbl

[33] H. Takahata, $L_{\infty}$ bounds for asymptotic normality of weakly dependent summands using Stein's methods, Ann. Prob., 9, 676-683 (1981). | MR | Zbl

[34] A.N. Tikhomirov, On the convergence rate in the central limit theorem for weakly dependent random variables, Th. Prob. Appl., 25, 790-809 (1980). | Zbl | MR

[35] R.L. Tweedie, Sufficient conditions for ergodicity and recurrence of Markov chains, Stoch. Proc. Appl., 3, 385-403 (1975). | MR | Zbl

[36] R.L. Tweedie, The existence of moments for stationnary Markov chains, J. Appl. Prob., 20, 191-196 (1983). | MR | Zbl

[37] F.W. Warner, Foundations of differentiable manifolds and Lie groups, Singer, MIT (1971). | MR | Zbl

[38] H. Whitney, Elementary structure of real algebraic varieties, Ann. Math., 66, 545-556 (1957). | MR | Zbl

[39] C.S. Withers, Conditions for linear processes to be strong mixing, Z. Wahr. verw. gebiete, 57, 481-494 (1981). | MR | Zbl | DOI

[40] R. Yokoyama, Moments bounds for stationnary mixing sequences, Z. Wahr. verw. gebiete, 52, 45-57 (1980). | MR | Zbl | DOI

[41] Y. Rozanov, Stationary random processes, Holden Day Series (1967). | MR | Zbl

[42] E. Becker, On the real spectrum of a ring and its applications to semi algebraic geometry, Bull. A.M.S., 15, 19-60 (1986). | MR | Zbl | DOI

[1] Ahmad, I. (1980). Non parametric estimation of the Matusita's measure of affinity between absolutely continuous distributions. Ann. Inst. Stat. Math. 32, 241-245. | MR | Zbl

[2] Brezis, H. (1983). Analyse fonctionnelle. Théorie et applications. Masson, Paris. | MR | Zbl

[3] Devroye, L. and Gyorfi, L. (1985). Non parametric density estimation. The L1 view. Wiley, New-York. | MR

[4] Devroye, L. and Penrod, C.S. (1984). Distribution-free lower bounds in density estimation. Ann. Stat. 12, 1250-1262. | MR | Zbl

[5] Doukhan, P. Leon, J., Portal, F. Principe d'invariance faible pour la mesure empirique d'une suite de variables aléatoires mélangeantes. (To appear). | MR | Zbl | DOI

[6] Gorodetskii, V.V. (1977). On the strong mixing property for linear sequences. Theory Prob. Appl. 22, 411-413. | Zbl

[7] Matusita, K. (1955). Decision rules based on the distance for the problem of fit, two samples and estimation. Ann. Math. Stat. 26, 631-640. | MR | Zbl

[8] Mokkadem, A. (1985). Le modèle non linéaire AR(1) général. Ergodicité et ergodicité géométrique. C.R.A.S., t.301, 889-892. | MR | Zbl

[9] Mokkadem, A. (1986). Sur le mélange d'un processus ARMA vectoriel. C.R.A.S., t. 303, 519-521. | MR | Zbl

[10] Mokkadem, A. (1985). Etude des risques des estimateurs à noyaux. C.R.A.S., t. 301, 447-450. | MR | Zbl

[11] Parzen, E. (1962). On estimation of a probability density function and the mode. Ann. Math. Stat. 33, 1065-1076. | MR | Zbl

[12] Pham, D. and Tran, L.T. (1986). Some mixing properties of time series models. Stoch. Proc. Appl. 19, 297-303. | MR | Zbl

[13] Rosenblatt, M. (1956). Remarks on some non parametric estimates of a density function. Ann. Math. Stat. 27, 832-835. | MR | Zbl

[14] Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci., U.S.A. 42, 43-47. | MR | Zbl | DOI

[15] Rosenblatt, M. (1979). Global measure of deviation of kernel and nearest neighbor density estimates. Lect. Notes Math. 757, 181-190. | MR | Zbl

[16] Rudin, W. (1966). Real and Complex Analysis. Mc Graw-Hill, New-York. | MR | Zbl

[17] Schwartz, L. (1966). Théorie des distributions. Hermann, Paris. | MR | Zbl

[18] Weil, A. (1965). L'intégration dans les groupes topologiques et ses applications. Hermann, Paris. | MR | Zbl | JFM

[19] Whithers, C.S. (1981). Conditions for linear processes to be strong mixing. Z. Wahr. verw. Gebiete, 57, 481-494. | MR | Zbl

[20] Yokoyama, R. (1980). Moment bounds for stationary mixing sequences. Z. Wahr. verw. Gebiete, 52, 45-57. | MR | Zbl | DOI

[1] I. A. Ahmad and P. E. Lin, A nonparametric estimation of the entropy for absolutely continuous distributions, IEEE Trans. Inf. Theory, Vol.I , T.22, 372-375, 1976. | MR | Zbl

[2] G. P. Basharin, On statistical estimate for the entropy of a sequence of independent random variables, Theory Prob. Appl., Vol. 4, 333-336, 1956. | MR | Zbl

[3] L. Birgé, Estimation de l'entropie d'une densité, exposé du Séminaire d'Orsay, Paris 1985.

[4] L. Devroye and L. Györfi, Nonparametric density estimation. The L1 view, Wiley, New-York, 1985. | MR | Zbl

[5] E. J. Dudewicz and E. C. Vander Meulen, Entropy-base tests of Uniformity, J. Amer. Stat. Ass., Vol. 76, N° 376, 967-974, 1981 . | MR | Zbl | DOI

[6] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge University Press, 1967. | JFM | Zbl

[7] K. Hutcheson and L. R. Shenton, Some moments of an estimate of Shannon's measure of information, Comm. Stat., Vol. 3, 89-94, 1974. | MR | Zbl

[8] A. Mokkadem, Thèse doctorat d'état, Université Paris XI, Orsay, 1987

[9] B. L. S. Prakasa Rao, Nonparametric Functional Estimation, Academic Press, 1983. | MR | Zbl

[10] M. Rosenblatt, A central limit theorem and a strong mixing condition, Proc. Nat. Acad. Sc., U. S. A., 42, 43-47, 1956. | MR | Zbl | DOI

[11] M. Rosenblatt, Remark on some non parametric estimates of a density function, Ann. Math. Stat., 27, 832-835, 1956. | MR | Zbl

[12] L. Schwartz, Theorie des Distributions, Hermann, Paris, 1966. | MR | Zbl

[13] O. Vasicek, A test for normality based on sample entropy, J. Roy. Stat. Soc., Ser. B, 38, 54-59, 1976. | MR | Zbl

[1] R.B. Ash (1967), Information theory. Intersciences publishers. | MR | Zbl

[2] J.P. Burg (1978), Maximum entropy spectral analysis. In modern spectrum analysis, ed. D.G. Childers, Wiley, New York.

[3] J.P. Burg (1978), A new analysis technique for time series data. In modern spectrum analysis, ed. D.G. Childers, Wiley, New York.

[4] S. Cambanis (1982), Complex symmetric stable variables and processes. In : Contribution to Statistics : Essays in Honour of Norman L. Johnson, pp. 63-79. Sen, P.K., Ed New York, North Holland. | MR | Zbl

[5] S. Cambanis and G. Miller (1981), Linear problems in $p$ th order and stable processes. SIAM J. Appl. Math. 41, pp. 43-69. | MR | Zbl | DOI

[6] S. Cambanis and A.R. Soltani (1984), Prediction of stable processes : spectral and moving average representations. Z. Wahr. verw. gebiete, 66, pp. 593-612. | MR | Zbl | DOI

[7] I. Csiszar (1975), $I$ -divergence geometry of probability distributions and minimization problems. Ann. Prob., vol. 3, n° 1, pp. 146-158. | MR | Zbl

[8] Y. Hosoya (1982), Harmonizable stable processes. Z. Wahr. verw. gebiete 60, pp. 517-533. | MR | Zbl | DOI

[9] M. Kanter (1979), Lower bound for non linear prediction error in moving average processes, Ann. Prob., 7, pp. 128-138. | MR | Zbl

[10] M. Pinsker (1964), Information and information stability of random variables and processes. Holden Day Series. | MR | Zbl

[11] Y. Rozanov (1967), Stationary random processes, Holden Day Series. | MR | Zbl

[12] W. Rudin (1974), Real and complex analysis, Mc Graw Hill. | MR | Zbl

[13] M. Schilder (1970), Some structure theorems for the symmetric stable laws, Ann. Math. Stat., 41, p. 412-421. | MR | Zbl

[14] C.E. Shannon (1948), The mathematical theory of communications, Bell System Technical Journal. | MR | Zbl

[15] L.A. Shepp, D. Slepian and A.D. Wyner (1980), On prediction of moving average processes, Bell system Technical Journal vol. 59, n° 3, pp. 367-415. | MR | Zbl | DOI

[16] T. Subba Rao and M.M. Gabr (1984), An introduction to bispectral analysis and bilinear time series models. Springer Verlag. | MR | Zbl