Moment conditions for almost sure convergence of weakly correlated random variables. (English) Zbl 0785.60018

Bryc, W.; Smolenski, W.

Proc. Am. Math. Soc. 119, No. 2, 629-635 (1993).

Let \(\{\xi_ k, k\in N\}\) be a real random sequence on a probability space \((\Omega,{\mathcal M},P)\). Define \[ \widetilde\rho(k)= \sup\bigl\{\text{corr}(V;W);\;V\in L_ 2({\mathcal F}_ S),\;W\in L_ 2({\mathcal F}_ T\bigr\}, \] where \({\mathcal F}_ A\) denotes the \(\sigma\)- field generated by \(\xi_ k\), \(k\in A\), and the supremum is taken over all finite subsets \(S\), \(T\in N\) such that \(\text{dist}(S,T)\geq k\). Further, let \(\widetilde r(k)=\sup\bigl\{\text{corr}(V,W)\bigr\}\), where the supremum is taken over all finite subsets \(S\), \(T\subset N\) such that \(\text{dist}(S,T)\geq k\) and over all linear combinations \(V\) of variables \(\{\xi_ k;\;k\in S\}\) and all linear combinations \(W\) of variables \(\{\xi_ k;\;k\in T\}\). The authors obtain the following two results:
(i) If \(\widetilde\rho(k)<1\) for some \(k\), and if \(E \xi_ j=0\), \(E \xi^ 2_ j=1\) for all \(j\), \(\sup_ j E| \xi_ j|^{2+\delta}<\infty\) for some \(\delta\) \((>0)\) and \(\sum a^ 2_ j<\infty\), then \(\sum a_ j\xi_ j\) converges almost surely.
(ii) If \(\widetilde r(k)<1\) for some \(k\), \(E \xi_ j=0\) for all \(j\) and \(\sum j^{-3/2} E \xi^ 2_ j<\infty\), then \(n^{-1} \sum^ n_ 1 \xi_ j\to 0\) almost surely.

Reviewer: Ken-ichi Yoshihara (Yokohama)

MSC:

60F15	Strong limit theorems
60E15	Inequalities; stochastic orderings

Keywords:

maximal correlation; almost surely convergent series; strong law of large numbers

References:

[1]	Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. · Zbl 0271.60009
[2]	Richard C. Bradley, On the spectral density and asymptotic normality of weakly dependent random fields, J. Theoret. Probab. 5 (1992), no. 2, 355 – 373. · Zbl 0787.60059
[3]	-, Equivalent mixing conditions for random fields, Technical Report No. 336, Center for Stochastic Processes, Univ. of North Carolina, Chapel Hill, 1990.
[4]	W. Bryc, Conditional expectation with respect to dependent \?-fields, Proceedings of the seventh conference on probability theory (Braşov, 1982) VNU Sci. Press, Utrecht, 1985, pp. 409 – 411.
[5]	Raymond Cheng, On the rate of strong mixing in stationary Gaussian random fields, Studia Math. 101 (1992), no. 2, 183 – 191. · Zbl 0809.60062
[6]	Jean-Dominique Deuschel and Daniel W. Stroock, Large deviations, Pure and Applied Mathematics, vol. 137, Academic Press, Inc., Boston, MA, 1989. · Zbl 0705.60029
[7]	William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. · Zbl 0077.12201
[8]	F. Móricz, Moment inequalities for the maximum of partial sums of random fields, Acta Sci. Math. (Szeged) 39 (1977), no. 3-4, 353 – 366. · Zbl 0363.60022
[9]	Magda Peligrad, The \?-quick version of the strong law for stationary \?-mixing sequences, Almost everywhere convergence (Columbus, OH, 1988) Academic Press, Boston, MA, 1989, pp. 335 – 348.
[10]	Marcel Riesz, Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires, Acta Math. 49 (1927), no. 3-4, 465 – 497 (French). · JFM 53.0259.03
[11]	William F. Stout, On convergence of \?-mixing sequences of random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 31 (1974/75), 69 – 70. · Zbl 0283.60026
[12]	P. J. Szabłowski, Stochastic approximation with dependent disturbances. I, Comput. Math. Appl. 13 (1987), no. 12, 951 – 972. , https://doi.org/10.1016/0898-1221(87)90067-8 P. J. Szabłowski, Generalized laws of large numbers and auxiliary results concerning stochastic approximation with dependent disturbances. II, Comput. Math. Appl. 13 (1987), no. 12, 973 – 987. · Zbl 0644.62084