an:00428656
Zbl 0776.60045
Rio, Emmanuel
Strong approximation for set-indexed partial sum processes via KMT constructions. I
EN
Ann. Probab. 21, No. 2, 759-790 (1993).
00014647
1993
j
60F17 62G99
central limit theorem; set-indexed process; invariance principle; metric entropy; random measure; strong approximation; Vapnik-Chervonenkis class
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 \textit{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.
Zbl 0563.60033; Zbl 0554.60037