Smooth rate of weak convergence of convex type positive finite measures


Let μ be a positive finite measure of mass c0 on [a,b]⊂R. For a fixed x0 ε{lunate} [a, b] and τ(X) = |x - x0|, let the probability measure ρ{variant} = c0-1μ ○ 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](f{hook}(x)- Σ k=0 n f{hook}(k)(x0) k!(x-x0)k)·μ(dx)| and |∝[a, b]f{hook} dμ - f{hook}(x0)|, f{hook} ε{lunate} Cn([a, b]), n ≥ 1 are obtained involving a power moment of μ and the first modulus of continuity of f{hook}(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. © 1989.

Publication Title

Journal of Mathematical Analysis and Applications