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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
Full Text:
##### References:
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