New Representation Theorems for Completely Monotone and Bernstein Functions with Convexity Properties on Their Measures
- Resource Type
- Authors
- Shen Shan; Hristo S. Sendov
- Source
- Journal of Theoretical Probability. 28:1689-1725
- Subject
- Statistics and Probability
Discrete mathematics
Convex analysis
Monotone polygon
General Mathematics
Bernstein inequalities
Monotonic function
Statistics, Probability and Uncertainty
Measure (mathematics)
Bernstein polynomial
Convexity
Bernstein's theorem on monotone functions
Mathematics
- Language
- ISSN
- 1572-9230
0894-9840
In this paper, we investigate a class of Bernstein functions and a class of completely monotone functions with intriguing applications in convex analysis. We derive representation theorems for Bernstein and completely monotone functions with a convexity condition on their measures. These representation theorems are variants of the classical Bernstein and Levy–Khintchine representation theorems. We show that the transformations that turn a Bernstein function into one having corresponding Levy measure with harmonically concave tail are the same as the transformations that transform a completely monotone function into one having harmonically convex measure.