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Strong approximation for set-indexed partial sum processes via KMT constructions. III. (English) Zbl 0930.60016
Summary: [For parts I and II see Ann. Probab. 21, No. 2, 759-790 (1993; Zbl 0776.60045) and ibid., No. 3, 1706-1727 (1993; Zbl 0779.60030).]
We generalize the results of J. Komlós, P. Major and G. Tusnády [Z. Wahrscheinlichkeitstheorie Verw. Geb. 32, 111-131 (1975; Zbl 0308.60029) and ibid. 34, 33-58 (1976; Zbl 0307.60045)] concerning the strong approximation of partial sums of independent and identically distributed random variables with a finite \(r\)th moment to the case when the parameter set is two-dimensional. The most striking result is that the rates of convergence are exactly the same as in the one-dimensional case.

MSC:
60F15 Strong limit theorems
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References:
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