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Classes of probability density functions having Laplace transforms with negative zeros and poles

Published online by Cambridge University Press:  01 July 2016

Ushio Sumita  and
Yasushi Masuda
Ushio Sumita*
Affiliation:
University of Rochester
Yasushi Masuda*
Affiliation:
University of Rochester
*
∗Postal address: William E. Simon Graduate School of Business Administration, University of Rochester, Rochester, NY 14627, USA.
∗Postal address: William E. Simon Graduate School of Business Administration, University of Rochester, Rochester, NY 14627, USA.
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Abstract

We consider a class of functions on [0,∞), denoted by Ω, having Laplace transforms with only negative zeros and poles. Of special interest is the class Ω+ of probability density functions in Ω. Simple and useful conditions are given for necessity and sufficiency of f ∊ Ω to be in Ω+. The class Ω+ contains many classes of great importance such as mixtures of n independent exponential random variables (CMn), sums of n independent exponential random variables (PF∗n), sums of two independent random variables, one in CMr and the other in PF∗1 (CMPFn with n = r + l) and sums of independent random variables in CMn(SCM). Characterization theorems for these classes are given in terms of zeros and poles of Laplace transforms. The prevalence of these classes in applied probability models of practical importance is demonstrated. In particular, sufficient conditions are given for complete monotonicity and unimodality of modified renewal densities.

Keywords

RATIONAL LAPLACE TRANSFORMSSUMS AND MIXTURES OF EXPONENTIAL RANDOM VARIABLESCHARACTERIZATION THEOREMSCOMPLETE MONOTONICITYUNIMODALITYMODIFIED RENEWAL DENSITIESBIRTH-DEATH PROCESSES
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 3 , September 1987 , pp. 632 - 651
Copyright
Copyright © Applied Probability Trust 1987 

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Footnotes

This paper is dedicated with respect to Julian Keilson in honor of his 60th birthday.

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