Trigonometric fractional approximation of stochastic processes

Here we encounter and study very general stochastic positive linear operators induced by general positive linear operators that are acting on continuous functions in the trigonometric sense. These are acting on the space of real fractionally dierentiable stochastic processes. Under some very mild, general and natural assumptions on the stochastic processes we pro-duce related trigonometric fractional stochastic Shisha-Mond type inequalities of Lq-type 1 q \u3c 1 and corresponding trigonometric fractional stochastic Korovkin type theorems. These are regarding the trigonometric stochastic q-mean fractional 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 trigonometric fractional stochastic inequalities involving the stochastic modulus of continuity of the α-th fractional derivatives of the engaged stochastic process, α \u3e 0, α ∉ N. The impressive fact is that only two basic real Korovkin test functions assumptions, one of them trigonometric, are enough for the con-clusions of our trigonometric fractional stochastic Korovkin theory. We give applications to stochastic Bernstein operators in the trigonometric sense