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Strong approximation for set-indexed partial-sum processes, via KMT constructions. II. (English) Zbl 0779.60030
Summary: [For part I see ibid. 21, No. 2, 759-790 (1993; Zbl 0776.60045).]
Let $$(X_ i)_{i \in \mathbb{Z}^ d_ +}$$ be an array of zero-mean independent identically distributed random vectors with values in $$\mathbb{R}^ k$$ with finite variance, and let $${\mathcal S}$$ be a class of Borel subsets of $$[0,1]^ d$$. If, for the usual metric, $${\mathcal S}$$ is totally bounded and has a convergent entropy integral, we obtain a strong invariance principle for an appropriately smoothed version of the partial-sum process $$\{\sum_{i \in \nu S}X_ i:S \in{\mathcal S}\}$$ with an error term depending only on $${\mathcal S}$$ and on the tail distribution of $$X_ 1$$. In particular, when $${\mathcal S}$$ is the class of subsets of $$[0,1]^ d$$ with $$\alpha$$-differentiable boundaries introduced by R. Dudley [J. Approximation Theory 10, 227-236 (1974; Zbl 0275.41011)], we prove that our result is optimal.

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