Smooth rate of weak convergence of convex type positive finite measures

Abstract

Let μ be a positive finite measure of mass c 0 on [ a, b]⊂R. For a fixed x 0 ϵ [ a, b] and τ(X) = ¦x − x 0¦ , let the probability measure ϱ = c 0 −1 μ ○ τ −1. Assume that the corresponding to distribution function fulfills certain higher-order convexity conditions. By the use of convex moment methods, upper bounds for |∫ [a,b](ƒ(x)− Σ k=0 n ƒ (k)(x 0) k! (x−x 0) k)·μ(dx)| and ¦∝ [a, b]ƒ dμ − ƒ(x 0)¦, ƒ ϵ C n([a, b]), n ⩾ 1 are obtained involving a power moment of μ and the first modulus of continuity of ƒ (n) . These produce sharp inequalities that are attained. The established estimates improve the corresponding ones in the literature. Applications to probabilistic distributions are given at the end.