Large Deviations and Idempotent Probability: 1st Edition (Hardback) book cover

Large Deviations and Idempotent Probability

1st Edition

By Anatolii Puhalskii

Chapman and Hall/CRC

520 pages | 10 B/W Illus.

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Hardback: 9781584881988
pub: 2001-05-07
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Description

In the view of many probabilists, author Anatolii Puhalskii's research results stand among the most significant achievements in the modern theory of large deviations. In fact, his work marked a turning point in the depth of our understanding of the connections between the large deviation principle (LDP) and well-known methods for establishing weak convergence results.

Large Deviations and Idempotent Probability expounds upon the recent methodology of building large deviation theory along the lines of weak convergence theory. The author develops an idempotent (or maxitive) probability theory, introduces idempotent analogues of martingales (maxingales), Wiener and Poisson processes, and Ito differential equations, and studies their properties. The large deviation principle for stochastic processes is formulated as a certain type of convergence of stochastic processes to idempotent processes. The author calls this large deviation convergence.

The approach to establishing large deviation convergence uses novel compactness arguments. Coupled with the power of stochastic calculus, this leads to very general results on large deviation asymptotics of semimartingales. Large and moderate deviation asymptotics are treated in a unified manner.

Starting with the foundations of idempotent measure theory and culminating in applications to large deviation asymptotics of queueing systems, Large Deviations and Idempotent Probability offers an outstanding opportunity to examine both the development of a remarkable approach and recently discovered results as presented by one of the foremost leaders in the field.

Reviews

"…an original and significant approach (completely elaborated) to the large deviation theory through possibility theory, which, as a result of this book, can be viewed as a large deviation limit of the probability theory."

-Mathematical Reviews Clippings, 2002

Table of Contents

IDEMPOTENT PROBABILITY THEORY

Idempotent Probability Measures

Idempotent Measures

Measurable Maps

Modes of Convergence

Idempotent Integration

Product Spaces

Independence and Conditioning

Idempotent Distributions and Laplace-Fenchel Transforms'

Idempotent Measures on Topological Spaces

Idemptent Measures on Projective Limits

Topological Spaces of Idempotent Probabilities

Maxingales

Stopping Times

Idempotent Stochastic Processes

Exponential Maxingales

Wiener and Poisson Idempotent Processes

Continuous Local Maxingales

Idempotent Ito Equations

Semimaxingales and Maxingale Problems

Proofs of the Uniqueness Results

Convergence of Idempotent Processes

LARGE DEVIATION CONVERGENCE

Large Deviation Convergence in Tihonov Spaces

General Theory

Large Deviation Convergence in the Skorohod Space

The Method of Finite-Dimensional Distributions

Convergence of Stochastic Exponentials

LD Convergence via Convergence of the Characteristics

Corollaries

The Method of the Maxingale Problem

Convergence of Stochastic Exponentials

Convergence of Characteristics

APPLICATIONS

Markov Processes

Queueing Networks

APPENDIX

About the Series

Monographs and Surveys in Pure and Applied Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
BUS049000
BUSINESS & ECONOMICS / Operations Research
MAT003000
MATHEMATICS / Applied
MAT029010
MATHEMATICS / Probability & Statistics / Bayesian Analysis