Smooth rate of weak convergence of convex type positive finite measures
Abstract
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
Recommended Citation
Anastassiou, G. (1989). Smooth rate of weak convergence of convex type positive finite measures. Journal of Mathematical Analysis and Applications, 141 (2), 491-508. https://doi.org/10.1016/0022-247X(89)90193-5
