We present a stochastic model predictive control (MPC) method for linear discrete-time systems subject to possibly unbounded and correlated additive stochastic disturbance sequences. Chance constraints are treated in analogy to robust MPC using the concept of probabilistic reachable sets for constraint tightening. We introduce an initialization of each MPC iteration which is always recursively feasibility and thereby allows that chance constraint satisfaction for the closed-loop system can readily be shown. Under an i.i.d. zero mean assumption on the additive disturbance, we furthermore provide an average asymptotic performance bound. Two examples illustrate the approach, highlighting feedback properties of the novel initialization scheme, as well as the inclusion of time-varying, correlated disturbances in a building control setting.