Convergence faible de la statistique linéaire de rang pour des variables aléatoires faiblement dépendantes et non stationnaires

Harel, Michel

Directeur de thèse : Raoult, Jean-Pierre

Membres du jury : Puri, Madan ; Raoult, Jean-Pierre ; Dacunha-Castelle, Didier ; Deheuvels, Paul ; Dolecki, Szymon ; Hallin, Marc ; Massart, Pascal ; Ruymgaart, Fritz

Thèses d'Orsay, no. 251 (1989) , 382 p.

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@phdthesis{BJHTUP11_1989__0251__P0_0,
     author = {Harel, Michel},
     title = {Convergence faible de la statistique lin\'eaire de rang pour des variables al\'eatoires faiblement d\'ependantes et non stationnaires},
     series = {Th\`eses d'Orsay},
     publisher = {Universit\'e Paris-Sud Centre d'Orsay},
     number = {251},
     year = {1989},
     language = {fr},
     url = {http://www.numdam.org/item/BJHTUP11_1989__0251__P0_0/}
}

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Harel, Michel. Convergence faible de la statistique linéaire de rang pour des variables aléatoires faiblement dépendantes et non stationnaires. Thèses d'Orsay, no. 251 (1989), 382 p. http://numdam.org/item/BJHTUP11_1989__0251__P0_0/

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2.bis Harel, M., Puri M. L. (1987), Convergence faible du processus empirique multidimensionnel corrigé en condition de mélange. C.R.A.S. 305, série 1, 93-95. | MR | Zbl

3. Harel, M., Puri, M. L. (1986), The space ${\tilde{D}}_{k}$ and weak convergence for the rectangle-indexed processes under mixing. Technical Report, Department of Mathematics, Indiana University, 40 p. | Zbl

3.bis Harel, M., Puri, M. L. (1988),L'espace Dk et la convergence faible du processus empirique indexé par rectangles en condition de mélange. C.R.A.S. t. 306, série I. 207-210. | MR | Zbl

4. Harel, M. (1984), Convergence en loi pour la topologie de Shorohod éclatée du processus empirique multidimensionnel normalisé tronqué éclaté et corrigé. Statistique et Analyse des données, n°2, Vol. 9, 68-91. | MR | Zbl | Numdam

5. Harel, M. (1988), Weak convergence of multidimensional rank statistics in $ϕ$ mixing condition. Journal of Statistical Planning and Inference, 20, 41-63. | MR | Zbl | DOI

5.bis Harel, M. (1986), Convergence faible de la statistique de rang multidimensionnelle en condition de $φ$ mélange C.R.A.S. t. 306, série I 433-436. | MR | Zbl

6. Harel, M. (1986), Convergence faible de la statistique de rang multidimensionnelle en condition de $φ$ mélange ou mélange fort. Cahiers du C.E.R.O. Vol. 28, n°1, 2, 3, 131-142. | MR | Zbl

7. Harel, M., Puri, M. L. (1989), Weak convergence of the serial linear rank statistic under mixing conditions with applications to time series and Markov processes. A paraître dans Annals of Probability. | MR | Zbl

7.bis Harel, M., Puri, M. L. (1987), Convergence faible de la statistique sérielle linéaire de rang en condition de dépendance avec application aux séries chronologiques et processus de Markov, C.R.A.S. t. 304, série I, 583-586. | MR | Zbl

8. Harel, M., Puri, M. L. (1989), Weak convergence of the serial linear rank statistic with unbounded scores and regression constants under mixing conditions. A paraître dans Journal of Statistical Planning and inference. | MR | Zbl

8.bis Harel, M., Puri, M. L. (1988), Convergence faible de la statistique sérielle linéaire de rang avec des fonctions de scores et des constantes de régression non bornées en condition de mélange C.R.A.S. t. 307, série I. | MR | Zbl

9. Harel, M., Puri, M. L. (1988), Weak convergence of the simple linear rank statistic for a large class of score functions under mixing conditions in the non stationary case. Technical report, Department of Mathematics, Indiana University, 24 p.

10. Harel, M., Puri, M. L. (1989), Limiting behavior of U-statistics, V-statistics and one sample rank order statistics for non stationary absolutely regular processes. Journal of Multivariate Analysis, 30, 181-204. | MR | Zbl

10.bis Harel, M., Puri, M. L. (1988), Comportement limite de la U-statistique, de la V-statistique et d'une statistique de rang à un échantillon pour des processus absolument réguliers non stationnaires. C.R.A.S. t. 306, série I, 625-628. | MR | Zbl

11. Harel, M., Puri, M. L. (1989), Weak convergence of the U-statistic and weak invariance of the one-sample rank order statistic for Markov processes and ARMA models. Journal of Multivariate Analysis, 31, 258-265. | MR | Zbl | DOI

12. Harel, M., Puri, M. L. (1989), Limiting behaviour of one sample rank order statistics with unbounded scores for non stationary absolutely regular processes. A paraître dans Journal of Statistical Planning and Inference. | MR | Zbl

13. Harel, M., Puri, M. L. (1989), Weak invariance of generalized U-statistics for non stationary absolutely regular processes. A paraître dans Stochastic Processes and their Applications | MR | Zbl

13.bis Harel, M., Puri, M. L. (1989), Convergence faible de la U-statistique généralisée pour des processus non stationnnaires absolument réguliers. C.R.A.S. t. 309, série I, 135-138 | MR | Zbl

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[2] L. Rüschendorf Asymptotic distributions of Multivariate Rank order Statistics Ann. of Stat. Vol. 4 N° 5 p. 912-923 (1976) | MR | Zbl | DOI

[3] T.R. Fears and K. Mehra Weak convergence of a two sample empirical processes and a Chernoff Savage theorem for $ϕ$ mixing sequence Ann. of Stat. Vol. 2 N° 3 p. 586-596 (1974) | MR | Zbl | DOI

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Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. | MR | Zbl

Doukhan, P. and Portal, F. (1983). Moment de variables aléatoires mélangeantes. C.R. Acad. Sc. Paris 297. 129-132. | MR | Zbl

Einmahl, J.H.J., Ruymgaart, F.H. and Wellner, J.A. (1984). Criteria for weak convergence of weighted multivariate empirical processes indexed by points or rectangles. Acta. Sci. Math. (Szeged).

Fears, T.R. and Mehra, K. (1974). Weak convergence of a two sample empirical process and a Chernoff-Savage theorem for $ϕ$ -mixing sequences. Ann Stat. 2, 586-596. | MR | Zbl | DOI

Harel, M. (1980). Convergence en loi pour la topologie de Skorohod du processus empirique multidimensionnel normalisé tronqué et semi-corrigé. Lecture Notes in Mathematics. Springer Verlag, 821. 46-85. | MR | Zbl

Mehra, K.L. and Rao, M.S. (1975). Weak convergence in dq-metrics of multidimensional empirical processes. Preprint, University of Alberta. Edmonton. Canada.

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Withers, C.S. (1975). Convergence of empirical processes of mixing rv's on [0,1]. Ann. Stat. 3, 1101-1108. | MR | Zbl | DOI

[1] I. Ahmad et P. E. Lin, On the Chernoff-Savage theorem for dependent random sequences, Ann. Inst. Stat. Math., 32, 1980, p. 211-222. | MR | Zbl | DOI

[2] S. Balacheff et G. Dupont, Normalité asymptotique des processus empiriques tronqués et des processus de rang, Lecture Notes in Mathematics, Springer Verlag, 821, 1980, p. 19-45. | MR | Zbl

[3] P. Doukhan et F. Portal, Moment de variables aléatoires mélangeantes, Comptes rendus, 297, série I, 1983, p. 129-132. | MR | Zbl

[4] J. H. J. Einmahl, F. H. Ruymgaart et J. A. Wellner, Criteria for weak convergence of weighted multivariate empirical processes indexed by points or rectangles, Acta. Sc. Math., Szeged, 1984.

[5] T. R. Fears et K. Mehra, Weak convergence of a two sample empirical process and a Chernoff-Savage theorem for $φ$ -mixing sequences, Ann. Stat., 2, 1974, p. 586-596. | MR | Zbl | DOI

[6] M. Harel (1980) : Convergence en loi pour la topologie de Skorohod du processus empirique multidimensionnel normalisé tronqué et semi-corrigé, Lecture Notes in Math., Springer Verlag, n° 821, 1980, p. 46-85. | MR | Zbl

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[8] C. S. Withers, Convergence of empirical processes of mixing rv's on [0,1], Ann. Stat., 3, 1975, p. 1101-1108. | MR | Zbl | DOI

Ahmad, I. and Lin, P.E. (1980). On the Chernoff-Savage theorem for dependent random sequences. Ann. Inst. Stat. Math. 32. 211-222. | MR | Zbl | DOI

Alexander, K.S. (1982). Some limit theorems for weighted and non-identically distributed empirical processes. Ph.D. Thesis, M.I.T. | MR

Balacheff, S. and Dupont, G. (1980). Normalité asymptotique des processus empiriques tronqués et des processus de rang. Lecture Notes in Mathematics. Springer Verlag 821. 19-45. | MR | Zbl

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Einmahl, J.H.J, Ruymgaart, F.H. and Wellner, J.A. (1984). Criteria for weak convergence of weighted multivariate empirical processes indexed by points or rectangles. Acta. Sci. Math. (Szeged).

Fears, T.R. and Mehra, K. (1974). Weak convergence of a two sample empirical process and a Chernoff-Savage theorem for $ϕ$ mixing sequence. Ann. of Stat. Vol. 2. No. 3. 586-596. | MR | Zbl | DOI

Neuhaus, G. (1971). On weak convergence of stochastic processes with multidimensional time parameters. Ann. Math. Statist. 42. 1285-1295. | MR | Zbl

Neuhaus, G. (1975). Convergence of the reduced empirical process for non i.i.d. random vectors. Ann. of Stat. 3. 528-531. | MR | Zbl | DOI

Neumann, N. (1982). Ein Schwaches Invarianzprinzip für den gewichteten empirischen Prozess von gleichmässig mischenden Zufallsvariablen. Ph.D. Thesis, Göttingen, W. Germany. | Zbl

Pyke, R, and Shorack, G.R. (1968). Weak convergence of a two sample empirical process and a new approach to Chernoff-Savage theorems. Ann. Math. Stat. 39. 755-771. | MR | Zbl | DOI

Rüschendorf, L. (1974). On the empirical process of multivariate, dependent random variables. Journal of Multivariate Analysis 4, 469-478. | MR | Zbl | DOI

Rüschendorf, L. (1976). Asymptotic distributions of multivariate rank order statistics. Ann. Stat. 4. 912-923. | MR | Zbl | DOI

Shorack, G.R. and Wellner, J.A. (1982). Limit theorems and inequalities for the uniform empirical process indexed by intervals. Ann. Probability 10. 639-652. | MR | Zbl

Ruymgaart, F.M. and Wellner, J.A. (1984). Some properties of weighted multivariate empirical processes. Stat. and Decisions 2, 199-223. | MR | Zbl

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Straf, M.L. (1972). Weak convergence of stochastic processes with several parameters. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 2. 187-221. Univ. of California Press. | MR | Zbl

Withers, C.S. (1975). Convergence of empirical processes of mixing rv's on [0,1]. Ann. Stat. 3. 1101-1108. | MR | Zbl

[1] S. Balacheff et G. Dupont, Normalité asymptotique des processus empiriques tronqués et des processus de rang, Lecture Notes in Mathematics, n° 821, Springer Verlag, 1980, p. 19-45. | MR | Zbl

[2] P. Billingsley, Convergence of probability Measures, Wiley, New York, 1968. | MR | Zbl

[3] J. H. J. Einmahl, F. H. Ruymgaart et J. A. Wellner, Criteria for weak convergence of weighted multivariate empirical processes indexed by points or rectangles, Acta. Sci. Math. (Szeged) 1988 (à paraître). | MR | Zbl

[4] M. Harel, Convergence en loi pour la topoiogie de Skorohod du processus empirique multidimensionnel normalisé tronqué et semi-corrigé, Lecture Notes in Mathematics, n° 821, Springer Verlag, 1980, p. 46-85. | MR | Zbl

[5] M. Harel et M. L. Puri, Weak convergence of Weighted Multivariate Empirical Processes under Mixing Conditions, Proceedings of the 6th Panonian Conference, Autriche, 1987. | Zbl

[6] G. Neuhaus, On weak convergence of stochastic processes with multidimensional time parameters, Ann. Math. Statist., 42, 1971, p. 1285-1295. | MR | Zbl

(1) S. Balacheff et G. Dupont Sur la convergence des suites de processus multidimensionnels normalisés tronqués et mélangeants. Thèse de 3ème Cycle Université de Rouen (1979)

(2) S. Balacheff et G. Dupont Normalité asymptotique des processus empiriques tronqués et des processus de rang. Lecture Notes in Mathematics Springer Verlag (1980), n° 821 pp. 19-45. | MR | Zbl

(3) P.J. Bickel and J. Wichura Convergence criteria for multiparameter stochastic processes and some applications. Ann. of Math. Stat. Vol. 42 n° 5 pp. 1656-1670 (1971) | MR | Zbl | DOI

(4) P. Billingsley Convergence of Probability Measures, Wiley (1968). | MR | Zbl

(5) R.M. Dudley Central Limit theorems for empirical measures. Ann. Probability, 6, pp. 899-929 (1978) | MR | Zbl

(6) M. Harel Convergence en loi pour la topologie de Skorohod du processus empirique multidimensionnel normalisé tronqué et semi-corrigé Lectures Notes in Mathematics, Springer-Verlag (1980), n° 821 pp. 46-85. | MR | Zbl

(7) L. Rüschendorf On the empirical Process of Multivariate, dependent Random Variables Journal of Multivariate Analysis 4, pp. 469-478 (1974) | MR | Zbl | DOI

Ahmad, I. and P.E. Lin (1980). On the Chernoff-Savage theorem for dependent random sequences. Ann. Inst. Stat. Math. 32, 211-222. | MR | Zbl | DOI

Balacheff, S. and G. Dupont (1980). Normalité asymptotique des processus empiriques tronqués et des processus de rang. Lecture Notes in Mathematics No. 821. Springer, Berlin-New York. | MR | Zbl

Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. | MR | Zbl

Denker, M. and U. Rösler (1985). Some contributions to Chernoff-Savage theorems. Statist. and Decision 3, 49-75. | MR | Zbl

Dudley, R.M. (1978). Central limit theorems for empirical measures. Ann. Probab. 6, 899-929. | MR | Zbl

Fears, T.R. and K. Mehra (1974). Weak convergence of two sample empirical processes and a Chernoff-Savage theorem for $ϕ$ mixing sequence. Ann. Stat. 2 (3), 586-596. | MR | Zbl | DOI

Harel, M. (1980). Convergence en loi pour la topologie de Skorohod du processus empirique multi-dimensionnel normalisé tronqué et semi-corrigé. Lecture Notes in Mathematics No. 821. Springer, Berlin-New York. | MR | Zbl

Harel, M. (1983). Convergence pour les processus empiriques éclatés. Ann. Sci. Univ. Clermont Ferrand II. | MR | Zbl | Numdam

Harel, M. (1984). Convergence en loi pour la topologie de Skorohod éclatée du processus empirique multidimensionnel normalisé tronqué éclaté et corrigé. Statist. et Analyse des Données 9 (2), 68-91. | MR | Zbl | Numdam

Harel, M. (1986). A generalization of a theorem of Van Zuijlen to the $ϕ$ mixing case. Technical report, U.A. C.N.R.S. 759, Rouen.

Harel, M. and M.L. Puri (1987). Weak convergence of weighted multivariate empirical processes under mixing conditions. Proceedings of the 6th Pannonian Symposium on Mathematical Statistics, Volume A: Mathematical Statistics and Probablity. Reidel, Dordrecht-Boston, 121-141. | MR | Zbl

Ibragimov, J.A. and Y.V. Linnik (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen. | MR | Zbl

Mehra, K.L. and M.S. Rao (1975). Weak convergence in dq-metrics of multidimensional empirical processes. Preprint, Univ. of Alberta, Edmonton, Canada.

Neumann, N. (1982). Ein Schwaches Invarianzprinzip für den gewichteten empirischen Prozess von gleichmässig mischenden Zufalsvariablen. Ph.D. Thesis, Göttingen. | Zbl

Pyke, R. and G. Shorack (1968). Weak convergence of two sample empirical processes and a new approach of Chernoff-Savage theorems. Ann. Math. Statist. 39, 755-771. | MR | Zbl | DOI

Rüschendorf, L. (1974). Of the empirical process of multivariate, dependent random variables. J. Multivariate Anal. 4, 469-478. | MR | Zbl | DOI

Rüschendorf, L. (1976). Asymptotic distributions of multivariate rank order statistics. Ann. Statist. 4 (5), 912-923. | MR | Zbl | DOI

Ruymgaart, F.H. (1974). Asymptotic normality of non-parametric tests for independence. Ann. Statist. 2 (5), 892-910. | MR | Zbl | DOI

Ruymgaart, F.H. and M.C. Van Zuijlen (1978). Asymptotic normality of multivariate linear rank statistics in the non-i.i.d. case. Ann. Statist. 6 (3), 588-602. | MR | Zbl | DOI

Van Zuijlen, M.C. (1976). Some properties of the empirical distribution function in the non-i.i.d. case. Ann. Statist. 4 (2), 406-408. | MR | Zbl

Withers, C.S. (1975). Convergence of empirical processes of mixing rv's on [0,1]. Ann. Statist. 3 (5), 1101-1108. | MR | Zbl | DOI

[1] S. Balacheff et G. Dupont, Normalité asymptotique des processus empiriques tronqués et de processus de rang, Lecture Notes in Mathematics, Springer Verlag, n° 821, 1980, p. 19-45. | MR | Zbl

[2] M. Harel, Convergence en loi pour la topologie de Skorohod du processus empirique multidimensionnel tronqué et semi-corrigé, Lecture Notes in Mathematics, Springer Verlag, n° 821, 1980, p. 46-85. | MR | Zbl

[3] M. Harel, Convergence en loi pour la topologie de Skorohod éclatée du processus empirique multidimensionnel tronqué éclaté et corrigé, Stat. et An. des données, 9, n° 2, 1984. | MR | Zbl | Numdam

Ahmad, I. and Lin, P.E. (1980). On the Chernoff-Savage theorem for dependent random sequences. Ann. Inst. Stat. Math. 32, 211-222. | MR | Zbl | DOI

Balacheff, S. and Dupont, G. (1980). Normalité asymptotique des processus empiriques tronqués et des processus de rang. Lectures Notes In Mathematics. Springer Verlag, n° 821. | MR | Zbl

Billingsley, P. (1968). Convergence probability measures. Wiley. | MR | Zbl

Denker, M. and Rosler, U. Preprint (1983). Some contributions to Chernoff-Savage theorems. (to appear in Statistical and Decision (1985)). | MR | Zbl

Fears, T.R. and Mehra, K. (1974). Weak convergence of a two sample empirical processes and a Chernoff-Savage theorem for $ϕ$ mixing sequence. Ann. of Stat. Vol. 2, n° 3, 586-596. | MR | Zbl | DOI

Harel, M. (1980). Convergence en loi pour la topologie de Shorohod du processus empirique multidimensionnel normalisé tronqué et semi-corrigé. Lectures Notes In Mathematics. Springer Verlag n° 821. | MR | Zbl

Harel, M. (1983). Convergence pour les processus empirique éclatés. Ann. Scient. de l'Université de Clermont Ferrand II. | MR | Zbl | Numdam

Harel, M. (1984). Convergence en loi pour la topologie de Shorohod éclatée du processus empirique multidimensionnel normalisé tronqué éclaté et corrigé. Statistique et Analyse des données, n° 2, vol. 9. | MR | Zbl | Numdam

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