Korovkin inequalities for stochastic processes


The rate of convergence of a sequence of positive linear stochastic operators {Tj}j ∈ n to the unit operator I on spaces of stochastic processes X is studied. This is mainly done for stochastic processes that are smooth over a compact and convex index set Q ⊂ Rk, k ≥ 1. Nearly best upper bounds are given for |E(TjX)(x0) - (EX)(x0)|, x0 ε{lunate} Q, where E is the expectation operator and Tj are E-commutative. These lead to strong and elegant inequalities many times sharp, which involve the first modulus of continuity of EXα, where Xα is a (partial) derivative of X. The case of Q being a compact convex subset of a real normed vector space is also met and there the upper bound is the best possible. © 1991.

Publication Title

Journal of Mathematical Analysis and Applications