Other Statistics and Probability Commons^™

Completely Monotone And Bernstein Functions With Convexity Properties On Their Measures, Shen Shan

Electronic Thesis and Dissertation Repository

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 …