SLLN and convergence rates for nearly orthogonal sequences of random variables

Let $\{ {X_k}:k \geqslant 1\}$ be a sequence of random variables with finite second moments $EX_k^2 = \sigma _k^2 < \infty$ for which $\vert E{X_k}{X_l}\vert \leqslant {\sigma _k}{\sigma _l}\rho (\vert k - l\vert)$, where $\{ \rho (j):j \geqslant 0\}$ is a sequence of nonnegative numbers with $\sum\nolimits_{j = 0}^\infty {\rho (j)} < \infty$. In particular, in the case of orthogonality, $\rho (j) = 0$ for $j \geqslant 1$. We prove strong laws for the first arithmetic means ${\zeta _n} = {n^{ - 1}}\sum\nolimits_{k = 1}^n {{X_k}}$ and the Cesàro means

$${\tau _n} = {n^{ - 1}}\sum\limits_{k = 1}^n {(1 - (k - 1){n^{ - 1}}){X_k}} .$$

The convergence rates are studied in the form $P\{ {\sup _{n > {2^p}}}\vert{\zeta _n}\vert > \varepsilon \}$ and $P\{ {\sup _{n > {2^p}}}\vert{\tau _n}\vert > \varepsilon \}$, where $\varepsilon > 0$ is fixed and $p$ tends to $\infty$. At the end, the case where $\Sigma _{j = 0}^\infty \rho (j) = \infty$ is also treated.