New Representation Theorems for Completely Monotone and Bernstein Functions with Convexity Properties on Their Measures

Hristo Sendov () and Shen Shan ()
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Hristo Sendov: Western University
Shen Shan: Western University

Journal of Theoretical Probability, 2015, vol. 28, issue 4, 1689-1725

Abstract: Abstract 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 Lévy–Khintchine representation theorems. We 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.

Keywords: Completely monotone function; Bernstein function; Lévy measure; Lévy–Khintchine representation; Bernstein representation; Harmonically convex (concave); Coupon collector’s problem; Primary 26A48; 26D07; Secondary 30E20 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10959-014-0557-9

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