In this paper, the problem of stability, recursive feasibility and convergence conditions of stochastic model predictive control for linear discrete-time systems affected by a large class of correlated disturbances is addressed. A stochastic model predictive control that guarantees convergence, average cost bound and chance constraint satisfaction is developed. The results rely on the computation of probabilistic reachable and invariant sets using the notion of correlation bound. This control algorithm results from a tractable deterministic optimal control problem with a cost function that upper-bounds the expected quadratic cost of the predicted state trajectory and control sequence. The proposed methodology only relies on the assumption of the existence of bounds on the mean and the covariance matrices of the disturbance sequence.