Strong limit theorems for blockwise -dependent and blockwise quasi-orthogonal sequences of random variables

Author:
F. Móricz

Journal:
Proc. Amer. Math. Soc. **101** (1987), 709-715

MSC:
Primary 60F15; Secondary 60G50

MathSciNet review:
911038

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a sequence of random variables with zero mean and finite variance . We say that is blockwise -dependent if for each large enough the following is true: if we remove or more consecutive 's from the dyadic block , then the two remaining portions are independent. We say that is blockwise quasiorthogonal if for each , the expectations are small in a certain sense again within the dyadic block . Blockwise independence and blockwise orthogonality are particular cases of the above notions, respectively.

We study the a.s. behavior of the series and that of the first arithmetic means . It turns out that the classical strong limit theorems, with one exception, remain valid in this more general setting, too.

**[1]**G. Alexits,*Convergence problems of orthogonal series*, Translated from the German by I. Földer. International Series of Monographs in Pure and Applied Mathematics, Vol. 20, Pergamon Press, New York-Oxford-Paris, 1961. MR**0218827****[2]**Wassily Hoeffding and Herbert Robbins,*The central limit theorem for dependent random variables*, Duke Math. J.**15**(1948), 773–780. MR**0026771****[3]**F. Móricz,*Moment inequalities and the strong laws of large numbers*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**35**(1976), no. 4, 299–314. MR**0407950****[4]**F. Móricz,*The strong laws of large numbers for quasi-stationary sequences*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**38**(1977), no. 3, 223–236. MR**0436294****[5]**Pál Révész,*The laws of large numbers*, Probability and Mathematical Statistics, Vol. 4, Academic Press, New York-London, 1968. MR**0245079****[6]**K. Tandori,*Bemerkungen zum Gesetz der großen Zahlen*, Period. Math. Hungar.**2**(1972), 33–39 (German). Collection of articles dedicated to the memory of Alfréd Rényi, I. MR**0339325**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
60F15,
60G50

Retrieve articles in all journals with MSC: 60F15, 60G50

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1987-0911038-3

Keywords:
-dependent random variables,
orthogonal random variables,
first arithmetic means,
strong limit theorems,
strong laws of large numbers

Article copyright:
© Copyright 1987
American Mathematical Society