In this paper, we research convergence of minimax approximation problems  of special class of bilevel stochastic programming. First, under regularity conditions of feasible set, we expand optimal solution set of lower level original stochastic programming to into non-singleton set. And we give continuity of optimal value and upper semi-convergence of the optimal solution  set on the upper level decision variables for lower level stochastic programming approximation problem. Furthermore, we feedback $\varepsilon$-optimal solution vector function provided by the lower level stochastic programming into the objective function of the upper level stochastic programming problems, and obtain the continuity of optimal value and the upper semi-convergence of optimal solution set with respect to the minimal information (m.i.) probability metric for upper level programming.