Large Deviations and Idempotent Probability

Large Deviations and Idempotent Probability

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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.show more

Product details

  • Hardback | 520 pages
  • 162.3 x 240.3 x 34.3mm | 975.38g
  • Taylor & Francis Inc
  • CRC Press Inc
  • Bosa Roca, United States
  • English
  • 10 black & white illustrations
  • 1584881984
  • 9781584881988

Review quote

"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, 2002show more

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 APPENDIXshow more