The Annals of Probability

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

Emmanuel Rio

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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 $(\sum_{i \in \nu S}X_i: S \in \mathscr{J})$ by a Gaussian partial sum process $(\sum_{i \in \nu S}Y_i: S \in \mathscr{J})$, uniformly over all sets in a certain Vapnik-Chervonenkis class $\mathscr{J}$ of subsets of $\lbrack 0, 1\rbrack^d$. 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 $N(0, \operatorname{Var} X_1)$-distributed random vectors may be constructed in such a way that, up to a power of $\log \nu$, $\sup_S \in \mathscr{J} |\sum_{i \in \nu S} (X_i - Y_i)| = O(\nu^{(d - 1)/2} \vee \nu^{d/r}) \mathrm{a.s.},$ for any Vapnik-Chervonenkis class $\mathscr{J}$ fulfilling the uniform Minkowsky condition. From a 1985 paper of Beck, it is straightforward to prove that such a result cannot be improved, when $\mathscr{J}$ is the class of Euclidean balls.

Article information

Ann. Probab., Volume 21, Number 2 (1993), 759-790.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 62G99: None of the above, but in this section

Central limit theorem set-indexed process partial sum process invariance principle Vapnik-Chervonenkis class metric entropy random measure


Rio, Emmanuel. Strong Approximation for Set-Indexed Partial Sum Processes Via KMT Constructions I. Ann. Probab. 21 (1993), no. 2, 759--790. doi:10.1214/aop/1176989266.

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See also

  • Part II: Emmanuel Rio. Strong Approximation for Set-Indexed Partial-Sum Processes, Via KMT Constructions II. Ann. Probab., Volume 21, Number 3 (1993), 1706--1727.