Contributions to the theory of stochastic convexity and stochastic majorization.

Persistent Link:
http://hdl.handle.net/10150/186817
Title:
Contributions to the theory of stochastic convexity and stochastic majorization.
Author:
Li, Haijun.
Issue Date:
1994
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
This dissertation presents some contributions to the theory of stochastic convexity and stochastic majorization. In the first part of this dissertation, we develop an operator-analytic approach to study temporal stochastic convexity and concavity of Markov processes. We obtain sufficient and necessary conditions for the process {X(t),t ∊ S} which imply that the expectation Ef(X(t)) is a monotone convex (concave) function of t whenever f is a monotone convex (concave) function. Our operator-analytic approach is quite powerful, but not as intuitive as sample path approaches used in other works. However, using it, we can obtain results that we could not obtain otherwise. In particular, we show that a result of Shaked and Shanthikumar is incorrect and we prove two alternative versions of it. In the second part of this dissertation, we discuss some applications of stochastic convexity. Using the F-monotonicity and F-convexity developed in the first part of this dissertation, we obtain relations among the several notions of stochastic convexity and stochastic majorization, and characterize the relationship between stochastic submodularity and stochastic rearrangement. These results generalize and extend several known results in the literature. Applications of our results to stochastic allocation problems are also discussed.
Type:
text; Dissertation-Reproduction (electronic)
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Mathematics; Graduate College
Degree Grantor:
University of Arizona
Committee Chair:
Shaked, Moshe

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleContributions to the theory of stochastic convexity and stochastic majorization.en_US
dc.creatorLi, Haijun.en_US
dc.contributor.authorLi, Haijun.en_US
dc.date.issued1994en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractThis dissertation presents some contributions to the theory of stochastic convexity and stochastic majorization. In the first part of this dissertation, we develop an operator-analytic approach to study temporal stochastic convexity and concavity of Markov processes. We obtain sufficient and necessary conditions for the process {X(t),t ∊ S} which imply that the expectation Ef(X(t)) is a monotone convex (concave) function of t whenever f is a monotone convex (concave) function. Our operator-analytic approach is quite powerful, but not as intuitive as sample path approaches used in other works. However, using it, we can obtain results that we could not obtain otherwise. In particular, we show that a result of Shaked and Shanthikumar is incorrect and we prove two alternative versions of it. In the second part of this dissertation, we discuss some applications of stochastic convexity. Using the F-monotonicity and F-convexity developed in the first part of this dissertation, we obtain relations among the several notions of stochastic convexity and stochastic majorization, and characterize the relationship between stochastic submodularity and stochastic rearrangement. These results generalize and extend several known results in the literature. Applications of our results to stochastic allocation problems are also discussed.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.chairShaked, Mosheen_US
dc.contributor.committeememberMyers, Donalden_US
dc.contributor.committeememberWright, Larryen_US
dc.identifier.proquest9502617en_US
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