The rate of weak convergence of convex type positive finite measures

Abstract

Let μ be a positive finite measure of mass m of the non-empty convex subset M of the real normed vector space ( V, ∥ · ∥). For a fixed x 0 ϵ M and τ( x) = ∥ x − x 0∥, let the probability measure ϱ = m −1 μ ∘ τ −1. Assume that the corresponding ϱ distribution function fulfills certain convexity conditions. By the use of convex moment methods best upper bounds of ¦∫ M f dμ − f(x 0)¦ are obtained for f an integrable real valued function on M and a given power moment of μ. These lead to sharp inequalities, i.e., attainable inequalities involving the first modulus of continuity of f. The established estimates improve the corresponding ones in the literature. These have wide applications to concave positive linear operators typically arising from well-known probabilistic distributions.