Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Strong limit theorems for blockwise $ m$-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 $ \{ {X_k}:k = 1,2, \ldots \} $ be a sequence of random variables with zero mean and finite variance $ \sigma _k^2$. We say that $ \{ {X_k}\} $ is blockwise $ m$-dependent if for each $ p$ large enough the following is true: if we remove $ m$ or more consecutive $ X$'s from the dyadic block $ \{ {X_{{2^{p - 1}} + 1}}, \ldots ,{X_{{2^p}}}\} $, then the two remaining portions are independent. We say that $ \{ {X_k}\} $ is blockwise quasiorthogonal if for each $ p$, the expectations $ E({X_k}{X_l})$ are small in a certain sense again within the dyadic block $ \{ {X_{{2^{p - 1}} + 1}}, \ldots ,{X_{{2^p}}}\} $. Blockwise independence and blockwise orthogonality are particular cases of the above notions, respectively.

We study the a.s. behavior of the series $ \sum\nolimits_{k = 1}^\infty {{X_k}} $ and that of the first arithmetic means $ (1/n)\sum\nolimits_{k = 1}^n {{X_k}} $. It turns out that the classical strong limit theorems, with one exception, remain valid in this more general setting, too.


References [Enhancements On Off] (What's this?)

  • [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

Similar Articles

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: https://doi.org/10.1090/S0002-9939-1987-0911038-3
Keywords: $ m$-dependent random variables, orthogonal random variables, first arithmetic means, strong limit theorems, strong laws of large numbers
Article copyright: © Copyright 1987 American Mathematical Society