Geodesic
Decrease rate of the probabilities of $\varepsilon$-deviations for the means of stationary processes
Matematičeskie zametki, Tome 64 (1998) no. 3, pp. 366-372.

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The asymptotic behavior as $n\to\infty$ of the normed sums $\sigma_n=n^{-1}\sum_{k=0}^{n-1}X_k$ for a stationary process $X=(X_n, n\in\mathbb Z)$ is studied. For a fixed $\varepsilon>0$, upper estimates for $\mathsf P\bigl(\sup_{k\ge n} |\sigma_k|\ge\varepsilon\bigr)$ as $n\to\infty$ are obtained. 
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V. F. Gaposhkin. Decrease rate of the probabilities of $\varepsilon$-deviations for the means of stationary processes. Matematičeskie zametki, Tome 64 (1998) no. 3, pp. 366-372. http://geodesic.mathdoc.fr/item/MZM_1998_64_3_a3/

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