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A strong law for weighted sums of i.i.d. random variables. (English) Zbl 0833.60031

Let X, X 1 ,X 2 ,,X n , be i.i.d. random variables with 𝐄X=0 and S n = i=1 n a in X n , where {a in } i=1(1)n is an array of constants. E.g., for the sequence {S n } of weighted sums a strong law is proved in the case sup n 1 n i=1 n |a in | q q<,q(1;], in the following manner:

𝐄|X| p <,p:=q q-1,implies1 nS n 0a.s.(Theorem1.1).

Also the case q=1 is investigated. Extensions to more general normalizing sequences are given and necessary and sufficient conditions are discussed. Several well-known results are contained, e.g. in the case q=, i.e. sup|a in |< and p=1, the result by B. D. Choi and S. H. Sung [Stochastic Anal. Appl. 5, 365-377 (1987; Zbl 0633.60049)].


MSC:
60F15Strong limit theorems
60G50Sums of independent random variables; random walks
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