New representation theorems for completely monotone and Bernstein functions with convexity properties on their measures
type: Item
instance of: scholarly article
title: New representation theorems for completely monotone and Bernstein functions with convexity properties on their measures
zbMATH Open document ID: 1334.26021
author: Hristo S. Sendov
author: Shen Shan
published in: Journal of Theoretical Probability
publication date: 2016-01-13T00:00:00Z
review text: A non-negative function \(f:(0,\infty)\to[0,\infty)\) is called completely monotone, if it is infinitely differentiable and \((-1)^nf^{(n)}(x)\geq 0\) for all \(x>0\) and \(n\geq 1\). A non-negative function \(g:(0,\infty)\to[0,\infty)\) is a Bernstein function, if it is infinitely differentiable and \((-1)^{n-1}g^{(n)}(x)\geq 0\) for all \(x>0\) and \(n\geq 1\). A measure \(\mu\) on \([0,\infty)\) is called harmonically convex if the function \(x\mapsto x\mu[0,x]\) is convex on \((0,\infty)\). A measure \(\nu\) on \((0,\infty)\) is said to have harmonically concave tail if the function \(x\mapsto x\nu(x,\infty)\) is concave on \((0,\infty)\). The main results of the paper are the following. { Theorem 1.} For any completely monotone function \(f(x)\) and a number \(\alpha\in(0,2/3]\), there exists unique harmonically convex measure \(\mu_\alpha\) on \([0,\infty)\), such that \[ f(x)=\int_{[0,\infty)}e^{-x^{1/\alpha}t}\mu_\alpha(dt). \] { Theorem 2.} For any Bernstein function \(g(x)\) and a number \(\alpha\in(0,2/3]\), there exists unique triple \((a,b,\nu_\alpha)\), such that \[ g(x)=a+bx^{1/\alpha}+\int_{(0,\infty)}(1-e^{-x^{1/\alpha}t})\nu_\alpha(dt), \] where \(a,b\geq 0\) are constants, and \(\nu_\alpha\) is a measure on \((0,\infty)\) with harmonically concave tail. The measure \(\nu_\alpha\) satisfies the integrability condition \[ \int_{(0,\infty)}(1\wedge t)\nu_\alpha(dt)<\infty. \] Finally, the authors show that the transformations that turn a Bernstein function into one having corresponding Lévy measure with harmonically concave tail are the same as the transformations that transform a completely monotone function into one having harmonically convex measure.
reviewed by: Szymon Glab
Mathematics Subject Classification ID: 26A48
Mathematics Subject Classification ID: 26D07
Mathematics Subject Classification ID: 30E20
zbMATH DE Number: 6529751
zbMATH Keywords: completely monotone function
zbMATH Keywords: Bernstein function
zbMATH Keywords: Lévy measure
zbMATH Keywords: Lévy-Khintchine representation
zbMATH Keywords: Bernstein representation
zbMATH Keywords: harmonically convex measure
zbMATH Keywords: measure with harmonically concave tail
zbMATH Keywords: coupon collector's problem
MaRDI profile type: Publication
full work available at URL: s10959-014-0557-9
OpenAlex ID: W2064492668
cites work: Vector-valued Laplace Transforms and Cauchy Problems
cites work: Potential theory of special subordinators and subordinate killed stable processes
cites work: Q5690490
cites work: Characterization of d.c. Functions in terms of quasidifferentials
cites work: The Lebesgue-Stieltjes Integral
cites work: All convex invariant functions of hermitian matrices
cites work: The coupon collector's problem revisited: asymptotics of the variance
cites work: Q3999383
cites work: Convex Analysis on the Hermitian Matrices
cites work: Inequalities: theory of majorization and its applications
cites work: Reciprocally convex functions
cites work: Convex Analysis
cites work: Bernstein functions. Theory and applications
cites work: The shape of the Borwein-Affleck-Girgensohn function generated by completely monotone and Bernstein functions
cites work: Q5846853
DOI: 10.1007/S10959-014-0557-9
Recommended article: New characterizations of completely monotone functions and Bernstein functions, a converse to Hausdorff's moment characterization theorem
Recommended article: Q3179419
Recommended article: On the Bernstein-Hausdorff-Widder conditions for completely monotone functions
Recommended article: A new condition for maximal monotonicity via representative functions
Recommended article: Extensions of Bernstein's theorem on absolutely monotonic functions
Recommended article: Represensibility of cones of monotone functions in weighted Lebesgue spaces and extrapolation of operators on these cones
Recommended article: Trace inequalities for completely monotone functions and Bernstein functions
Recommended article: Q3422510
Recommended article: Some problems about the representation of monotone operators by convex functions
Recommended article: Q3450748
description: scientific article; zbMATH DE number 6529751