The absolute convergence of weighted sums of dependent sequences of random variables

Summary The sum a_nX_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 in 0<1, ¦a_n¦ < and E¦X_n¦ < ; if a_n=zⁿ 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_nzⁿ (all ¦z¦<1) then implies E(log¦X_n¦)₊ < , while if the X_n are non-negative stable r.v.s. of index , ¦a_nX_n¦< if and only if ¦a_n¦ < .