Quantitative Multivariate Random Korovkin Theory

We introduce and study the very general multivariate stochastic positive linear operators induced by general multivariate positive linear operators that are acting on multivariate continuous functions. These are acting on the spaces of real differentiable multivariate time stochastic processes. Under some very mild, general and natural assumptions on the stochastic processes we produce related multidimensional stochastic Shisha-Mond type inequalities of L-type and corresponding multidimensional stochastic Korovkin type theorems. These are regarding the stochastic 1-mean convergence of a sequence of multivariate 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 maximum of the multivariate stochastic moduli of continuity of the nth order partial derivatives of the engaged stochastic process, n ≥ 0. The astonishing fact here is that basic real Korovkin test functions assumptions are enough for the conclusions of our multidimensional stochastic Korovkin theory. We finish with an application. We are motivated by [2], [3], [8], [9]. 1. Auxiliary Results Definition 1. Let Q be a compact convex subset of Rk, k > 1. Let X(t, ω) be a stochastic process from Q × (Ω,B, P ) into R, where (Ω,B, P ) is a probability space. We define the q-mean multivariate first moduli of continuity of X by Ω1(X, δ)Lq := sup {(∫ Ω |X(x, ω)−X(y, ω)|qP (dω) )1/q : x, y ∈ Q, ‖x− y‖`1 ≤ δ} , δ > 0, 1 ≤ q < ∞. (1.1) 253