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A generalization of the Men’shov-Rademacher theorem. (English. Russian original) Zbl 1181.60043
Math. Notes 86, No. 6, 861-872 (2009); translation from Mat. Zametki 86, No. 6, 925-937 (2009).
Summary: For a sequence \(\{X_n\}_{n\geq 1}\) of random variables with finite second moment and a sequence \(\{b_n\}_{n\geq 1}\) of positive constants, new sufficient conditions for the almost sure convergence of \(\Sigma _{n\geq 1} X_n /b_n\) are obtained and the strong law of large numbers, which states that lim\(_{n \rightarrow \infty} \sum_{k=1}^n X_k /b_n = 0\) almost surely, is proved. The results are shown to be optimal in a number of cases. In the theorems, assumptions have the form of conditions on \(\rho _n = \text{sup}_k\) (E\(X_k X_{k+n})^{+}\), \[ r_n = \underset {k} {\text{sup}} \frac {(\text EX_k X_{k+n})^+}{(\text EX_k^2)^{1/2}(\text EX_{k+n}^2)^{1/2}}, \] \(\text EX_n^2\), and \(b_n\), where \(x^+ = x\vee 0\) and \(n \in \mathbb N\).

MSC:
60F15 Strong limit theorems
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