Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T17:56:20.428Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximations fortes pour des processus bivariés

Published online by Cambridge University Press:  20 November 2018

Nathalie Castelle
Nathalie Castelle*
Affiliation:
Laboratoire de Mathématiques—UMR 8628, Bât. 425, Université de Paris-Sud, 91405 Orsay Cedex, France, email: [email protected]
Rights & Permissions [Opens in a new window]

Résumé

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Nous établissons un résultat d’approximation forte pour des processus bivariés ayant une partie gaussienne et une partie empirique. Ce résultat apporte un nouveau point de vue sur deux théorèmes hongrois bidimensionnels établis précédemment, concernant l’approximation par un processus de Kiefer d’un processus empirique uniforme unidimensionnel et l’approximation par un pont brownien bidimensionnel d’un processus empirique uniforme bidimensionnel. Nous les enrichissons un peu et montrons que sous leur nouvelle forme ils ne sont que deux énoncés d’un même résultat.

Abstract

Abstract

We establish a strong approximation result for bivariate processes containing a Gaussian part and an empirical part. This result leads to a new point of view on two Hungarian bidimensional theorems previously established, about the approximation of an unidimensional uniform empirical process by a Kiefer process and the approximation of a bidimensional uniform empirical process by a bidimensional Brownian bridge. We enrich them slightly and we prove that, under their new fashion, they are but two statements of the same result.

Keywords

60F1760G1562G30
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 3 , 01 June 2002 , pp. 533 - 553
Copyright
Copyright © Canadian Mathematical Society 2002

References

Références

[1] Bennett, G., Probability inequalities for the sum of independants random variables. J. Amer. Statist. Assoc. 57 (1962), 33–45.Google Scholar
[2] Bretagnolle, J. et Massart, P., Hungarian constructions from the non asymptotic view point. Ann. Probab. 17 (1989), 239–256.Google Scholar
[3] Castelle, N. et Laurent-Bonvalot, F., Strong approximations of bivariate uniform empirical processes. Ann. Inst. Henri Poincaré Prob. Stat. 34 (1998), 425–480.Google Scholar
[4] Csörgʺo, M. et Revesz, P., Strong approximations in probability and statistics. Academic Press, New York, 1981.Google Scholar
[5] Csörgʺo, M. et Horváth, L., Weighted approximations in probability and statistics. Wiley & Sons, 1993.Google Scholar
[6] Komlós, J., Major, P. et Tusnády, G., An approximation of partial sums of independent RV′ - and the sample D. F. I. Z.Warsch. verw. Gebiete 32 (1975), 111–131.Google Scholar
[7] Mason, D. et van Zwet, R., A refinement of the KMT inequality for the uniform empirical process. Ann. Probab. 15 (1987), 871–884.Google Scholar
[8] Shorack, G. R. et Wellner, J. A., Empirical processes with applications to statistics. Wiley & Sons, 1986.Google Scholar
[9] Skorohod, A. V., On a representation of random variables. Theory. Probab. Appl. 21 (1976), 628–632.Google Scholar
[10] Tusnády, G., A remark on the approximation of the sample D.F. in the multidimensional case. Period. Math. Hungar. 8 (1977), 53–55.Google Scholar
[11] Tusnády, G., A Study of Statistical Hypotheses. Dissertation, The Hungarian Academy of Sciences, Budapest, 1977.Google Scholar
[12] Wellner, J., Limit theorems for the ratio of the empirical distribution function to the true distribution function. Z.Warsch. verw. Gebiete. 45 (1978), 73–88.Google Scholar