Moment conditions for almost sure convergence of weakly correlated random variables

Bryc, W.; Smoleński, W.

doi:10.1090/S0002-9939-1993-1149969-7

Moment conditions for almost sure convergence of weakly correlated random variables
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by W. Bryc and W. Smoleński

Proc. Amer. Math. Soc. 119 (1993), 629-635

DOI: https://doi.org/10.1090/S0002-9939-1993-1149969-7

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Abstract:

For random sequences with unrestricted maximal correlation coefficient strictly less than $1$, sufficient moment conditions for almost sure convergence of a series and for the strong law of large numbers are given.

References

Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
Richard C. Bradley, On the spectral density and asymptotic normality of weakly dependent random fields, J. Theoret. Probab. 5 (1992), no. 2, 355–373. MR 1157990, DOI 10.1007/BF01046741

Equivalent mixing conditions for random fields

W. Bryc, Conditional expectation with respect to dependent $\sigma$-fields, Proceedings of the seventh conference on probability theory (Braşov, 1982) VNU Sci. Press, Utrecht, 1985, pp. 409–411. MR 867452
Raymond Cheng, On the rate of strong mixing in stationary Gaussian random fields, Studia Math. 101 (1992), no. 2, 183–191. MR 1149571, DOI 10.4064/sm-101-2-183-191
Jean-Dominique Deuschel and Daniel W. Stroock, Large deviations, Pure and Applied Mathematics, vol. 137, Academic Press, Inc., Boston, MA, 1989. MR 997938
William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
F. Móricz, Moment inequalities for the maximum of partial sums of random fields, Acta Sci. Math. (Szeged) 39 (1977), no. 3-4, 353–366. MR 458535
Magda Peligrad, The $r$-quick version of the strong law for stationary $\varphi$-mixing sequences, Almost everywhere convergence (Columbus, OH, 1988) Academic Press, Boston, MA, 1989, pp. 335–348. MR 1035254
Marcel Riesz, Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires, Acta Math. 49 (1927), no. 3-4, 465–497 (French). MR 1555250, DOI 10.1007/BF02564121
William F. Stout, On convergence of $\phi$-mixing sequences of random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 31 (1974/75), 69–70. MR 358941, DOI 10.1007/BF00538717
P. J. Szabłowski, Stochastic approximation with dependent disturbances. I, Comput. Math. Appl. 13 (1987), no. 12, 951–972. MR 898943, DOI 10.1016/0898-1221(87)90067-8