Upper and lower bounds of large deviations for some dependent sequences

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Let $${\{X_{n}, n\geq1\}}$$ be a sequence of random variables with $${S_n=\sum_{i=1}^nX_i}$$ and $${M_n=\max \{X_1,X_2,\ldots, X_n\}}$$ . Under some suitable conditions, we establish the upper bound of large deviations for $${S_n}$$ and $${M_n}$$ based on some dependent sequences including acceptable random variables, widely acceptable random variables and a class of random variables that satisfies the Marcinkiewicz–Zygmund type inequality and Rosenthal type inequality. In addition, the lower bound of large deviations for some dependent sequences is also obtained. The results obtained in the paper generalize and improve some corresponding ones for independent random variables and negatively associated random variables.