Sur la loi des grands nombres pour les martingales vectorielles et l'estimateur des moindres carrés d'un modèle de régression
Duflo, M. ; Senoussi, R. ; Touati, A.
Annales de l'I.H.P. Probabilités et statistiques, Tome 26 (1990), p. 549-566 / Harvested from Numdam
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@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},
     volume = {26},
     year = {1990},
     pages = {549-566},
     mrnumber = {1080585},
     zbl = {0722.60031},
     language = {fr},
     url = {http://dml.mathdoc.fr/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) pp. 549-566. http://gdmltest.u-ga.fr/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