Random central limit theorems (CLTs) are established for a linear process driven by a strictly stationary ${psi}$-weakly dependent process as well as for the ${psi}$-weakly dependent process itself, whose dependence structure was introduced by Doukhan and Louhichi [Doukhan, P., & Louhichi, S. (1999). A new weak dependence condition and applications to moment inequalities. Stochastic Processes and their Applications, 30 84, 313-342] to generalize mixings and other dependence. Random CLTs are established for partial sums and sample autocovariances of the ${psi}$-weakly dependent process and the linear process under absolute summability.