Multivariate stochastic Korovkin theory given quantitatively

Abstract

We introduce and study 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 space 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 Lq-type 1 ≤ q < ∞ and corresponding multidimensional stochastic Korovkin type theorems. These are regarding the stochastic q-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 give an application. © 2007 Elsevier Ltd. All rights reserved.