Summary
As a generalization of a theorem of Chow [1] it is shown by an elementary method that for i.i.d. r.v.'s X 1,...,X n, with expectation zero and finite p-th absolute moment (p≧2) the weighted sums \(\sum\limits_{i = 1}^n {a_{n,i} X_i /n^{1/p} (\sum\limits_{i = 1}^n {a_{n,i}^2 } } )^{1/2} \) converge to zero a.s.
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Thrum, R. A remark on almost sure convergence of weighted sums. Probab. Th. Rel. Fields 75, 425–430 (1987). https://doi.org/10.1007/BF00318709
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DOI: https://doi.org/10.1007/BF00318709