#### Degree

Doctor of Philosophy

#### Program

Statistics and Actuarial Sciences

Hristo S. Sendov

#### Abstract

The concepts of completely monotone and Bernstein functions have been introduced near one hundred years ago. They find wide applications in areas ranging from stochastic L\'{e}vy processes and complex analysis to monotone operator theory. They have well-known Bernstein and L\'{e}vy-Khintchine integral representations through which there are one-to-one correspondences between them and Radon measures on $[0,\infty)$ or $(0,\infty)$, respectively. In this thesis, we investigate subclasses of completely monotone and Bernstein functions with various convexity properties on their measures. These subclasses have intriguing applications in probability theories and convex analysis.

The convexity properties we investigate include convexity, harmonic convexity and $\beta$-convexity of the cumulative distribution functions. We characterize measures with various convexity properties to obtain results analogous to the classical P\'{o}lya's Theorem. Then we apply these characterizations of the measures to derive integral representations for these classes of completely monotone and Bernstein functions that are variants of the classical Bernstein and L\'{e}vy-Khintchine integral representations.

To explore the connections among completely monotone and Bernstein functions with various convexity properties on their measures, we investigate the characterizations and obtain various necessary and sufficient conditions for a completely monotone or Bernstein function to belong to one of the subclasses. We also identify maps that transform completely monotone and Bernstein functions into one with certain convexity properties on their measures. Interesting parallels between completely monotone and Bernstein functions are observed. For example, the transformation that turn a Bernstein function into one having L\'{e}vy measure with harmonically concave tail is the same as the transformation that turns a completely monotone function into one having harmonically convex measure. To help understand these analogies, a criteria for completely monotone and Bernstein function to have measures with $\beta$-convexity property is obtained.That generalizes the conditions for both convexity and harmonic convexity.

Let $\mathcal{H}_{CM}$ be the set of all Bernstein functions $h$, such that $f\circ h$ is the Laplace transform of a harmonically convex measure for {\it any} completely monotone function $f$. Similarly, let $\mathcal{H}_{BF}$ be the set of all Bernstein functions $h$, such that $g\circ h$ has L\'{e}vy measure with harmonically concave tail for {\it any} Bernstein function $g$. Surprisingly, we show that $\mathcal{H}_{CM} = \mathcal{H}_{BF}$ and are non-empty. For example we prove that $x^\alpha$ is in $\mathcal{H}_{BF}$ for any $\alpha \in (0, {2}/{3}]$. In other words, the Bernstein function $x \mapsto x^\alpha$ is a transformation that deforms the measure of any Bernstein (resp. completely monotone) function into one that not only has a continuous distribution function on $(0,\infty)$ but also a convenient concavity (reps. convexity) property. We give necessary and sufficient condition for a Bernstein function to be in $\mathcal{H}_{BF}$ in terms of its convolution semigroups of sub-probability measures. However, it is not well-understood what are the functions that generate'' this set. We hope to investigate such issues in the future.

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