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Almost-sure results for a class of dependent random variables. (English) Zbl 0928.60025

Authors’ abstract: “The main of this note is to establish almost sure Marcinkiewicz-Zygmund type results for a class of random variables indexed by ${ℤ}_{+}^{d}$ – the positive $d$-dimensional lattice points – and having maximal coefficient of correlation strictly smaller than 1. The class of applications include filters of certain Gaussian sequences and Markov processes.”

The authors get in particular a strong law of large numbers similar to N. Etemadi’s one [Z. Wahrscheinlichkeitstheorie Verw. Geb. 55, 119-122 (1981; Zbl 0438.60027)], but under a weaker condition: they assume only that the maximal coefficient of correlation is $<1$, instead of pairwise independence in Etemadi’s theorem. A corresponding statement is also given for $d$-dimensional random fields (Theorem 6).

##### MSC:
 60F15 Strong limit theorems 60G60 Random fields