Strong approximation for set-indexed partial-sum processes, via KMT constructions. II. (English) Zbl 0779.60030

Rio, Emmanuel

Ann. Probab. 21, No. 3, 1706-1727 (1993).

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

Keywords:

central limit theorem; set-indexed process; invariance principle; metric entropy with inclusion; multivariate empirical processes; strong invariance principle