Sur la loi des grands nombres pour les martingales vectorielles et l'estimateur des moindres carrés d'un modèle de régression
Annales de l'I.H.P. Probabilités et statistiques, Tome 26 (1990) no. 4, p. 549-566
@article{AIHPB_1990__26_4_549_0,
     author = {Duflo, Michel and Senoussi, R. and Touati, A.},
     title = {Sur la loi des grands nombres pour les martingales vectorielles et l'estimateur des moindres carr\'es d'un mod\`ele de r\'egression},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {26},
     number = {4},
     year = {1990},
     pages = {549-566},
     zbl = {0722.60031},
     mrnumber = {1080585},
     language = {fr},
     url = {http://http://www.numdam.org/item/AIHPB_1990__26_4_549_0}
}
Duflo, M.; Senoussi, R.; Touati, A. Sur la loi des grands nombres pour les martingales vectorielles et l'estimateur des moindres carrés d'un modèle de régression. Annales de l'I.H.P. Probabilités et statistiques, Tome 26 (1990) no. 4, pp. 549-566. http://www.numdam.org/item/AIHPB_1990__26_4_549_0/

[1] T.W. Anderson et J. Taylor, Strong Consistency of Least Squares Estimators in Dynamic Models, Ann. Stat., vol. 7, 1979, p. 484-489. | MR 527484 | Zbl 0407.62040

[2] Y.S. Chow, A Martingale Inequality and the Law of Large Numbers, Proc. Am. Math. Soc., vol. 11, 1960, p. 107-111. | MR 112190 | Zbl 0102.13501

[3] Y.S. Chow, Local Convergence of Martingales and the Law of Large Numbers, Ann. Math. Stat., vol. 36, 1965, p. 493-507. | MR 182040 | Zbl 0134.34003

[4] N. Christopeit, Quasi-least Squares Estimation in Semi-Martingale Regression Models, Stochastics, vol. 16, 1986, p. 255-278. | MR 837614 | Zbl 0586.62137

[5] A.R. Darwich, Une loi du logarithme itéré pour les martingales locales multidimensionnelles et son application en régression linéaire stochastique, C. R. Acad. Sci. Paris, série I, 1989, p. 387-390. | MR 1054258 | Zbl 0721.62029

[6] D. Dacunha-Castelle et M. Duflo, Probabilités et statistiques, tome 2, Masson, 1983. | MR 732786 | Zbl 0535.62004

[7] M. Duflo, R. Senoussi et A. Touati, Propriétés asymptotiques presque sûres de l'estimateur des moindres carrés d'un modèle autorégressif vectoriel, Ann. Inst. Henri Poincaré, série B, vol. 27, n° 1, 1991. | Numdam | Zbl 0741.62083

[8] W.A. Fuller et D.P. Hasza, Properties of Predictors for Autoregressive Time Series, J. Am. Stat. Assoc., vol. 76, 1981, p. 155-161. | MR 608187 | Zbl 0465.62093

[9] A.J. Heunis, Asymptotic Properties of Prediction Error Estimators in Approximate System Identification, Stochastic, vol. 24, 1988, p. 1-43. | MR 966645 | Zbl 0648.60048

[10] H. Kaufmann, On the Strong Law of Large Numbers for Multivariate Martingales, Stoch. Proc. Appl., vol. 26, 1987, p. 73-85. | MR 917247 | Zbl 0632.60040

[11] P.R. Kumar, Convergence of Adaptive Control Schemes using Least Squares Parameter Estimates, I.E.E.E. Trans. Aut. Control., 1990, (à paraître). | Zbl 0716.93078

[12] T.L. Lai, Asymptotically Efficient Adaptive Control in Stochastic Regression Models, Adv. Appl. Math., vol. 7, 1986, p. 23-45. | MR 834218 | Zbl 0615.93041

[13] T.L. Lai et H. Robbins, Adaptive Design and Stochastic Approximation, Ann. Stat., vol. 7, n° 6, 1979, p. 1196-1221. | MR 550144 | Zbl 0426.62059

[14] T.L. Lai et H. Robbins, Consistency and Asymptotic Efficiency of Slope Estimates in Stochastic Approximation Schemes, Z. Wahr. Verw. Gebiete, vol. 56, 1981, p. 329- 360. | MR 621117 | Zbl 0472.62089

[15] T.L. Lai, H. Robbins et C.Z. Wei, Strong Consistency of Least Squares Estimates in Multiple Regression, II, J. Mult. Anal., vol. 9, 1979, p. 343-361. | MR 548786 | Zbl 0416.62051

[16] T.L. Tai et C.Z. Wei, Least Squares Estimates in Stochastic Regression Models with Applications to Identification and Control of Dynamic Systems, Ann. Math. Stat., vol. 10, 1982, p. 154-166. | MR 642726 | Zbl 0649.62060

[17] T.L. Lai et C.Z. Wei, Asymptotic Properties of Projections with Applications to Stochastic Regression Problems, J. Mult. Anal., vol. 12, 1982, p. 346-370. | MR 666011 | Zbl 0501.62083

[18] T.L. Tai et C.Z. Wei, Asymptotic Properties of General Autoregressive Models and Strong Consistency of Least Squares Estimates and their Parameters, J. Mult. Anal., vol. 13, 1983, p. 1-23. | MR 695924 | Zbl 0509.62081

[19] T.L. Tai et C.Z. Wei, Asymptotic Properties of Multivariate Weighted Sums with Applications to Stochastic Regression in Linear Dynamic Systems, Multivariate analysis, VI, éd. P. R. KRISHNAIAH, North-Holland, 1985, p. 375-393. | MR 822308 | Zbl 0607.62120

[20] A. Le Breton et M. Musiela, Consistency Sets of Least Squares Estimates in Stochastic Regression Models, Stoch. diff. systems, Lect. Notes Control Inf. Sci., vol. 126, 1989. | MR 1236070 | Zbl 0679.62055

[21] A. Le Breton et M. Musiela, Laws of Large Number for Semimartingales with Applications to Stochastic Regression, Probab. Theor Rel. Fields, vol. 81, 1989, p. 275- 290. | MR 982658 | Zbl 0662.60043

[22] D. Lépingle, Sur le comportement asymptotique des martingales locales, Lect. Notes Prob., vol. 649, Springer, 1978. | Numdam | MR 520004 | Zbl 0375.60062

[23] P. Lévy, Sur les séries dont les termes sont des variables indépendantes, Studia Math., vol. 3, 1931, p. 119-155. | JFM 57.0616.01 | Zbl 0003.30301

[24] A.V. Melnikov, The Law of Large Numbers for Multidimensional Martingales, Soviet Math. Dokl., vol. 33, 1986, p. 131-135. | MR 834691 | Zbl 0605.60047

[25] J. Neveu, Bases mathématiques du calcul des probabilités, Masson, 1964 (2e edition 1970). | MR 198504 | Zbl 0137.11203

[26] H. Robbins et D. Siegmund, A Convergence Theorem for Non Negative Almost Supermatingales and some Applications, dans J. S. RUSTAGI, Optimization methods in statistics, Academic Press, 1971, p. 233-257. | MR 343355 | Zbl 0286.60025

[27] V. Solo, Topics in Advanced Time Series Analysis, Lect. Notes Math., vol. 1215, Springer-Verlag, 1982. | MR 875627 | Zbl 0601.62108

[28] W. Stout, Almost Sure Convergence, Academic Press. | Zbl 0321.60022

[29] W. Stout, A Martingale Analogue of Kolmogorov's Law of Iterated Logarithm, Z. Wahrscheinlichkeitstheorie, vol. 15, 1970, p. 279-290. | MR 293701 | Zbl 0209.49004

[30] C.Z. Wei, Asymptotic Properties of Least Squares Estimates in Stochastic Regression Models, Ann. Stat., vol. 13, 1985, p. 1498-1508. | MR 811506 | Zbl 0582.62062

[31] C.Z. Wei, Adaptive Prediction by Least Squares Predictors in Stochastic Regression Models with Applications to Time Series, Ann. Stat., vol. 15, 1987, p. 1667-1682. | MR 913581 | Zbl 0643.62058