Stochastic Korovkin Theory given Quantitatively

We introduce and study very general stochastic positive linear operators induced by general positive linear operators that are acting on continuous functions. These are acting on the space of real differentiable stochastic processes. Under some very mild, general and natural assumptions on the stochastic processes we produce related stochastic Shisha–Mond type inequalities of L-type 1 ≤ q < ∞ and corresponding stochastic Korovkin type theorems. These are regarding the stochastic q-mean convergence of a sequence of stochastic positive linear operators to the stochastic unit operator for various cases. All convergences are produced with rates and are given via the stochastic inequalities involving the stochastic modulus of continuity of the n−th derivative of the engaged stochastic process, n ≥ 0. The impressive fact is that the basic real Korovkin test functions assumptions are enough for the conclusions of our stochastic Korovkin theory. We give an application.