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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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