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Strong approximations of bivariate uniform empirical processes. (English) Zbl 0915.60048

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.

MSC:

60F15 Strong limit theorems
60G15 Gaussian processes
62G30 Order statistics; empirical distribution functions
62H99 Multivariate analysis

Citations:

Zbl 0308.60029
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