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Abstract In this paper, we assume that the sequence {Yn,−∞<n<+∞} $\{Y_{n},-\infty< n<+\infty\}$ is a ρ− $\rho^{-}$-mixing random variables which are stochastically dominated by a random variable Y. Moreover, a real number sequence {an,−∞<n<+∞} $\{a_{n},-\infty< n<+\infty\}$ is assumed to be absolute summable. Then, complete convergence and complete γ-order moment convergence of the maximum partial sums for the moving average process {Xn=∑j=−∞+∞ajYn+j,n≥1} $\{X_{n}=\sum_{j=-\infty}^{+\infty}a_{j}Y_{n+j},n\geq1\}$ are obtained. The results in this paper extend and improve the corresponding ones under NA and ρ-mixing conditions in the literature.