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Strong approximation for set-indexed partial sum processes via KMT constructions. I. (English) Zbl 0776.60045
Summary: Let $$(X_ i)_{i\in\mathbb{Z}^ d_ +}$$ be an array of independent identically distributed zero-mean random vectors with values in $$\mathbb{R}^ k$$. When $$E(| X_ 1|^ r)<+\infty$$, for some $$r>2$$, we obtain the strong approximation of the partial sum process $$(\sum_{i\in\nu S}X_ i: S\in{\mathcal S})$$ by a Gaussian partial sum process $$(\sum_{i\in\nu S}Y_ i: S\in{\mathcal S})$$, uniformly over all sets in a certain Vapnik-Chervonenkis class $${\mathcal S}$$ of subsets of $$[0,1]^ d$$. The most striking result is that both an array $$(X_ i)_{i\in\mathbb{Z}^ d_ +}$$ of i.i.d. random vectors and an array $$(Y_ i)_{i\in\mathbb{Z}^ d_ +}$$ of independent $$N(0,\text{Var} X_ 1)$$-distributed random vectors may be constructed in such a way that, up to a power of $$\log\nu$$, $\sup_{S\in{\mathcal S}}\left|\sum_{i\in\nu S}(X_ i- Y_ i)\right|=O(\nu^{(d-1)/2}\lor\nu^{d/r})\;\text{ a.s.} ,$ for any Vapnik-Chervonenkis class $${\mathcal S}$$ fulfilling the uniform Minkowsky condition.
From a paper of J. Beck [Z. Wahrscheinlichkeitstheorie Verw. Geb. 70, 289-306 (1985; Zbl 0554.60037)], it is straightforward to prove that such a result cannot be improved, when $${\mathcal S}$$ is the class of Euclidean balls.

##### MSC:
 60F17 Functional limit theorems; invariance principles 62G99 Nonparametric inference
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