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Strong approximation for multivariate empirical and related processes, via KMT constructions. (English) Zbl 0675.60026
From author’s summary: Let P be the Lebesgue measure on the unit cube in \({\mathbb{R}}^ d\) and \(Z_ n\) be the centered and normalized empirical process associated with n independent observations with common law P. Given a collection of Borel sets \({\mathcal S}\) in \({\mathbb{R}}^ d\), it is known since R. M. Dudley’s work [J. Funct. Anal. 1, 290-330 (1967; Zbl 0188.205); J. Approximation Theory 10, 227-236 (1974; Zbl 0275.41011); Ann. Probab. 6, 899-929 (1978; Zbl 0404.60016); correction ibid. 7, 909-911 (1979)); Lect. Notes Math. 1097, 1-142 (1984; Zbl 0554.60029)] that if \({\mathcal S}\) is not too large then \(Z_ n\) may be strongly approximated by some sequence of Brownian bridges indexed by \({\mathcal S}\), uniformly over \({\mathcal S}\) with some rate \(b_ n.\)
We apply the one-dimensional dyadic scheme previously used by J. Komlós, P. Major and G. Tusnády (KMT) [Z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 111-131 (1975; Zbl 0308.60029); ibid. 34, 33-58 (1976; Zbl 0307.60045)] to get as good rates of approximation as possible in the above general multidimensional situation. The most striking result is that, up to a possible power of log(n), \(b_ n\) may be taken as \(n^{-1/2d}\) which is the best possible rate, when \({\mathcal S}\) is the class of Euclidean balls (this is the KMT result when \(d=1\), and the lower bounds are due to J. Beck [ibid. 70, 289-306 (1985; Zbl 0554.60037)] when \(d\geq 2)\).
Reviewer: P.Revesz

MSC:
60F17 Functional limit theorems; invariance principles
60G50 Sums of independent random variables; random walks
60G60 Random fields
62E20 Asymptotic distribution theory in statistics
62G30 Order statistics; empirical distribution functions
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