A discrete time infinite horizon stochastic program is considered whose objective function is a sum of discounted expected costs of decisions at time t. It is supposed that a cost function is convex, the number of outcomes is finite and the decision at time t is effected only by t and the decision at time t-1. The statistical model is essentially nonstationary. The boundedness assumption on the objective function is as weak as possible. A sequence of solvable finite horizon problems is defined and the convergence of their values and decisions to the value and decision of the infinite problem is proved. The results are refined when the underlying stochastic process is a finite homogeneous Markov chain.