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Starke Approximationen für durch Mengen indizierte Partialsummenprozesse und empirische Prozesse von mischenden Random Fields. (Strong approximations for set-indexed partial-sum processes and empirical processes of mixing random fields). (German) Zbl 0618.60033

Mathematische Fakultät der Albert-Ludwigs-Universität in Freiburg im Breisgau. 96 S. (1986).
Let \((X_ n\), \(j\in {\mathbb{N}}^ q)\) be a weakly dependent stationary random field of Banach space valued random elements, assumed to be strongly mixing in case the state space is finite-dimensional, and absolutely regular in the infinite-dimensional case. We obtain almost sure approximations of the partial-sum process \((\sum_{j\in nA}X_ j\), \(A\in {\mathcal A}\), \(n\in {\mathbb{N}})\) by a partial-sum process \((\sum_{j\in nA}Y_ j\), \(A\in {\mathcal A}\), \(n\in {\mathbb{N}})\) uniformly over all sets A in a certain class \({\mathcal A}\) of subsets of the q-dimensional unit cube. Here \((Y_ j\), \(j\in {\mathbb{N}}^ q)\) are i.i.d. Gaussian random vectors. These results are then applied to obtain the approximation of empirical processes over sets and indexed by sets by Gaussian partial sum processes indexed by sets.

MSC:

60F17 Functional limit theorems; invariance principles
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60G99 Stochastic processes