Castelle, Nathalie; Laurent-Bonvalot, Françoise Strong approximations of bivariate uniform empirical processes. (English) Zbl 0915.60048 Ann. Inst. Henri Poincaré, Probab. Stat. 34, No. 4, 425-480 (1998). Authors’ abstract: J. Komlós, P. Major and G. Tusnády [Z. Wahrscheinlichkeitstheorie Verw. Geb. 32, 111-131 (1975; Zbl 0308.60029)] constructed a strong approximation of the uniform empirical process \(\{\alpha_n(t), n\geq 1, t\in[0,1]\}\) by a Gaussian Kiefer process. We show that the global error bound provided by Komlós, Major and Tusnády may be improved by considering only local approximation. Moreover we provide explicit constants. We also prove a local refinement for Tusnády’s Gaussian strong approximation of the bidimensional uniform empirical process. The main technical tool we use is a non asymptotic normal approximation of the hypergeometric distribution. Reviewer: B.Le Gac (Marseille) Cited in 9 Documents MSC: 60F15 Strong limit theorems 60G15 Gaussian processes 62G30 Order statistics; empirical distribution functions 62H99 Multivariate analysis Keywords:strong approximation; bivariate empirical process; Brownian bridges; Gaussian Kiefer process; hypergeometric distribution Citations:Zbl 0308.60029 PDFBibTeX XMLCite \textit{N. Castelle} and \textit{F. Laurent-Bonvalot}, Ann. Inst. Henri Poincaré, Probab. Stat. 34, No. 4, 425--480 (1998; Zbl 0915.60048) Full Text: DOI Numdam EuDML