Measure-valued differentiation for finite products of measures : theory and applications

H. Leahu

Research output: ThesisPhd Thesis 4 Research NOT TU/e / Graduation NOT TU/e)

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Abstract

In this dissertation we perform a comprehensive analysis of measure-valued differentiation, in which weak differentiation of parameter-dependent probability measures plays a central role. We develop a theory of weak differentiation of measures and show that classical concepts such as differential calculus and analyticity extend to measure-valued mappings in some appropriate sense. Concurrently, we develop applications in the area of gradient estimation, perturbation analysis and Taylor series approximations for performance measures of complex stochastic systems. The key observation is that the weak derivative of some probability measure can be represented as the re-scaled difference between two probability measures, which leads to efficient simulating algorithms. More specifically, we show how weak derivatives can be used to construct (asymptotically) unbiased estimators for stochastic gradients or functional dependence of the performance measures of some stochastic systems, e.g., queueing networks or stochastic activity networks, with respect to some intrinsic parameter of the system. Eventually, we illustrate how weak derivatives can be used to derive Lipschitz constants for measure-valued mappings, which prove to be useful in performing perturbation analysis for the systems under consideration. A remarkable result shows that the stationary distribution of the waiting times in a G/G/1 queue is norm-continuous, with respect to some appropriate norm on the space of measures, provided that the service time distribution is weakly differentiable.
Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • Vrije Universiteit Amsterdam
Supervisors/Advisors
  • Tijms, Henk C., Promotor, External person
  • Heidergott, Bernd F., Copromotor, External person
Award date22 Sep 2008
Place of PublicationAmsterdam
Publisher
Print ISBNs978-90-5170-905-6
Publication statusPublished - 2008

Fingerprint

Measure Theory
Probability Measure
Perturbation Analysis
Stochastic Systems
Derivative
Performance Measures
Gradient Estimation
Norm
Stochastic Gradient
Differential Calculus
Queueing Networks
Unbiased estimator
Taylor series
Analyticity
Stationary Distribution
Waiting Time
Differentiable
Queue
Lipschitz
Complex Systems

Cite this

Leahu, H. (2008). Measure-valued differentiation for finite products of measures : theory and applications. Amsterdam: Vrije Universiteit Amsterdam.
Leahu, H.. / Measure-valued differentiation for finite products of measures : theory and applications. Amsterdam : Vrije Universiteit Amsterdam, 2008.
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author = "H. Leahu",
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publisher = "Vrije Universiteit Amsterdam",
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Leahu, H 2008, 'Measure-valued differentiation for finite products of measures : theory and applications', Doctor of Philosophy, Vrije Universiteit Amsterdam, Amsterdam.

Measure-valued differentiation for finite products of measures : theory and applications. / Leahu, H.

Amsterdam : Vrije Universiteit Amsterdam, 2008.

Research output: ThesisPhd Thesis 4 Research NOT TU/e / Graduation NOT TU/e)

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AB - In this dissertation we perform a comprehensive analysis of measure-valued differentiation, in which weak differentiation of parameter-dependent probability measures plays a central role. We develop a theory of weak differentiation of measures and show that classical concepts such as differential calculus and analyticity extend to measure-valued mappings in some appropriate sense. Concurrently, we develop applications in the area of gradient estimation, perturbation analysis and Taylor series approximations for performance measures of complex stochastic systems. The key observation is that the weak derivative of some probability measure can be represented as the re-scaled difference between two probability measures, which leads to efficient simulating algorithms. More specifically, we show how weak derivatives can be used to construct (asymptotically) unbiased estimators for stochastic gradients or functional dependence of the performance measures of some stochastic systems, e.g., queueing networks or stochastic activity networks, with respect to some intrinsic parameter of the system. Eventually, we illustrate how weak derivatives can be used to derive Lipschitz constants for measure-valued mappings, which prove to be useful in performing perturbation analysis for the systems under consideration. A remarkable result shows that the stationary distribution of the waiting times in a G/G/1 queue is norm-continuous, with respect to some appropriate norm on the space of measures, provided that the service time distribution is weakly differentiable.

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