An estimate of deviation probabilities of the sample mean of variables with \(\psi\)-semimixing (Q1897903)

For the probability \(P\left(\sum^n_{j= 1} X_j\geq na\right)\) \((a\in \mathbb{R}^1)\) of a strictly stationary random variable \(\{X_j, j\in Z\}\) with \(\psi\)-semimixing the Chernoff theorem is extended. Analogues of the theorems of \textit{D. Blackwell} and \textit{J. L. Hodges jun.} [Ann. Math. Stat. 30, 1113-1120 (1959; Zbl 0099.351)] are also proved.