1st Edition

Large Deviations and Idempotent Probability

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ISBN 9781584881988
Published May 7, 2001 by Chapman and Hall/CRC
520 Pages - 10 B/W Illustrations

USD $185.00

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Book 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.

Table of Contents

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
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 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
The Method of the Maxingale Problem
Convergence of Stochastic Exponentials
Convergence of Characteristics
Markov Processes
Queueing Networks

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Puhalskii\, Anatolii


"…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