SLLN and convergence rates for nearly orthogonal sequences of random variables
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- Proc. Amer. Math. Soc. 95 (1985), 287-294 Request permission
Abstract:
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 $|E{X_k}{X_l}| \leqslant {\sigma _k}{\sigma _l}\rho (|k - l|)$, 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 $P\{ {\sup _{n > {2^p}}}|{\zeta _n}| > \varepsilon \}$ and $P\{ {\sup _{n > {2^p}}}|{\tau _n}| > \varepsilon \}$, 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.References
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Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 95 (1985), 287-294
- MSC: Primary 60F15; Secondary 60G48
- DOI: https://doi.org/10.1090/S0002-9939-1985-0801340-6
- MathSciNet review: 801340