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Zbl 0857.60021
Chandra, Tapas K.; Ghosal, Subhashis
The strong law of large numbers for weighted averages under dependence assumptions. (English)
[J] J. Theor. Probab. 9, No.3, 797-809 (1996). ISSN 0894-9840; ISSN 1572-9230/e

The authors prove strong laws of large numbers for weighted averages of dependent random variables, generalizing the classical work of {\it B. Jamison}, {\it S. Orey} and {\it 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 $$P(X_i>s, X_j>t) - P(X_i>s) P(X_j>t) \le q \bigl(|i-j |\bigr) \alpha_{ij} (s,t), $$ together with a similar condition on $P(X_i<s, X_j<t)$. This condition generalizes the notion of asymptotic quadrant independence, introduced by {\it 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 The proofs make heavy use of unpublished results by the same authors.
[H.Dehling (Groningen)]

MSC 2000:: *60F15 Strong limit theorems

Keywords: law of large numbers; summability methods; weak dependence

Citations: Zbl 0141.16404; Zbl 0661.60048

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