A Generalization of the Menshov–Rademacher Theorem

For a sequence $\{X_n\}_{n\ge1}$ of random variables with finite second moment and a sequence $\{b_n\}_{n\ge1}$ of positive constants, new sufficient conditions for the almost sure convergence of $\sum_{n\ge1}X_n/b_n$ are obtained and the strong law of large numbers, which states that $\lim_{n\to\infty}\sum_{k=1}^nX_k/b_n=0$ 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 $\rho_n=\sup_k(\mathsf EX_kX_{k+n})^+$,
$$r_n=\sup_k\frac{(\mathsf EX_kX_{k+n})^+}{(\mathsf EX_k^2)^{1/2}(\mathsf EX_{k+n}^2)^{1/2}},$$
$\mathsf EX_n^2$, and $b_n$, where $x^+=x\vee0$ and $n\in\mathbb N$.