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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 \[ P(X_i>s, X_j>t) - P(X_i>s) P(X_j>t) \leq q \bigl(|i-j |\bigr) \alpha_{ij} (s,t), \] together with a similar condition on \(P(X_i<s, X_j<t)\). This condition generalizes the notion of asymptotic quadrant independence, introduced by 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 The proofs make heavy use of unpublished results by the same authors.

60F15 Strong limit theorems
Full Text: DOI
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