Summary
The sum ∑ a n X n of a weighted series of a sequence {X n } of identically distributed (not necessarily independent) random variables (r.v.s.) is a.s. absolutely convergent if for some α in 0<α≦1, ∑¦a n ¦α < ∞ and E¦X n ¦α < ∞; if a n =z n for some ¦z¦<1 then it suffices that E(log¦X n ¦)+<∞. Examples show that these sufficient conditions are not necessary. For mutually independent {X n } necessary conditions can be given: the a.s. absolute convergence of ∑ X n z n (all ¦z¦<1) then implies E(log¦X n ¦)+ < ∞, while if the X n are non-negative stable r.v.s. of index α, ∑¦a n X n ¦<∞ if and only if ∑¦a n ¦α < ∞.
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Daley, D.J. The absolute convergence of weighted sums of dependent sequences of random variables. Z. Wahrscheinlichkeitstheorie verw. Gebiete 58, 199–203 (1981). https://doi.org/10.1007/BF00531561
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DOI: https://doi.org/10.1007/BF00531561