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The strong law of large numbers for weighted averages under dependence assumptions. (English) Zbl 0857.60021

The authors prove strong laws of large numbers for weighted averages of dependent random variables, generalizing the classical work of B. Jamison, S. Orey and W. Pruitt [Z. Wahrscheinlichkeitstheorie Verw. Geb. 4, 40-44 (1965; Zbl 0141.16404)] for i.i.d. sequences. The dependence structure imposed is asymptotic quadrant sub-independent, requiring that

$P\left({X}_{i}>s,{X}_{j}>t\right)-P\left({X}_{i}>s\right)P\left({X}_{j}>t\right)\le q\left(|i-j|\right){\alpha }_{ij}\left(s,t\right),$

together with a similar condition on $P\left({X}_{i}. This condition generalizes the notion of asymptotic quadrant independence, introduced by T. Birkel [Stat. Probab. Lett. 7, No. 1, 17-20 (1988; Zbl 0661.60048)]. The authors also prove a Marcinkiewicz-Zygmund SLLN for weighted averages. The proofs make heavy use of unpublished results by the same authors.

MSC:
 60F15 Strong limit theorems
References:
 [1] Birkel, T. (1992). Laws of large numbers under dependence assumptions.Statist. Prob. Lett. 14, 355–362. · Zbl 0925.60023 · doi:10.1016/0167-7152(92)90096-N [2] Chandra, T. K. (1991). Extensions of Rajchman’s strong law of large numbers.Sankhyā, Ser. A 53, 118–121. [3] Chandra, T. K., and Ghosal, S. (1993). Some elementary strong laws of large numbers: a review. Technical Report #22/93, Indian Statistical Institute. [4] Chandra, T. K., and Ghosal, S. (1996). Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund.Acta Math. Hung. 72(3) (to appear). [5] Etemadi, N. (1983). Stability of sums of weighted random variables.J. Multivariate Anal. 13, 361–365. · Zbl 0531.60034 · doi:10.1016/0047-259X(83)90032-5 [6] Hall, P., and Heyde, C. C. (1980).Martingale Limit Theory and Its Application., Academic Press, New York. [7] Jamison, B., Orey, S., and Pruitt, W. E. (1965). Convergence of weighted averages of independent random variables.Z. Wahrsch. Verw. Gebiete 4, 40–44. · Zbl 0141.16404 · doi:10.1007/BF00535481 [8] McLeish, D. L. (1975). A maximal inequality and dependent strong laws.Ann. Prob. 3, 829–839. · Zbl 0353.60035 · doi:10.1214/aop/1176996269 [9] Pruitt, W. E. (1966). Summability of independent random variables.J. Math. Mech. 15, 769–776. [10] Rothatgi, V. K. (1971). Convergence of weighted sums of independent random variables.Proc. Cambridge Phil. Soc. 69, 305–307. · doi:10.1017/S0305004100046685 [11] Rosalsky, A. (1987). Strong stability of normed sums of pairwise i.i.d. random variables.Bull. Inst. Math. Acad. Sinica 15, 203–219.