In this technical note, the stability for time-varying discrete-time stochastic linear systems, with possibly unbounded trajectories, is associated to the existence of the long-run average cost criteria. Under controllability and observability, the stochastic system is stable in the sense that its state tends asymptotically to the origin in the mean. Further conditions related to a lower bound on the long-run average cost, or with periodic time-varying systems, provides uniform second moment stability.