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Strong Approximation for Set-Indexed Partial-Sum Processes, Via KMT Constructions II

Emmanuel Rio
The Annals of Probability
Vol. 21, No. 3 (Jul., 1993), pp. 1706-1727
Stable URL: http://www.jstor.org/stable/2244595
Page Count: 22
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Strong Approximation for Set-Indexed Partial-Sum Processes, Via KMT Constructions II
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Abstract

Let (Xi)i∈Zd + be an array of zero-mean independent identically distributed random vectors with values in Rk with finite variance, and let L be a class of Borel subsets of [ 0, 1]d. If, for the usual metric, L 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 {∑i∈ν S Xi: S ∈ L} with an error term depending only on L and on the tail distribution of X1. In particular, when L is the class of subsets of [ 0, 1]d with α-differentiable boundaries introduced by Dudley, we prove that our result is optimal.

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