Korovkin inequalities for stochastic processes

Abstract

The rate of convergence of a sequence of positive linear stochastic operators { T j } 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 ⊂ R k, k ⩾ 1. Nearly best upper bounds are given for ¦E(T jX)(x 0) − (EX)(x 0)¦, x 0 ϵ Q , where E is the expectation operator and T j are E-commutative. These lead to strong and elegant inequalities many times sharp, which involve the first modulus of continuity of EX α, where X α 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.