Decrease rate of the probabilities of $\varepsilon$-deviations for the means of stationary processes

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.