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

Author

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  • Hristo Sendov

    (Western University)

  • Shen Shan

    (Western University)

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.

Suggested Citation

  • Hristo Sendov & Shen Shan, 2015. "New Representation Theorems for Completely Monotone and Bernstein Functions with Convexity Properties on Their Measures," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1689-1725, December.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:4:d:10.1007_s10959-014-0557-9
    DOI: 10.1007/s10959-014-0557-9
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    References listed on IDEAS

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    1. Renming Song & Zoran Vondraček, 2006. "Potential Theory of Special Subordinators and Subordinate Killed Stable Processes," Journal of Theoretical Probability, Springer, vol. 19(4), pages 817-847, December.
    2. Hristo S. Sendov & Ričardas Zitikis, 2014. "The Shape of the Borwein–Affleck–Girgensohn Function Generated by Completely Monotone and Bernstein Functions," Journal of Optimization Theory and Applications, Springer, vol. 160(1), pages 67-89, January.
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