General Theory

  • Amarjit BudhirajaEmail author
  • Paul Dupuis
Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 94)


Throughout this chapter \(\{X^{n}\}_{n\in \mathbb {N}}\) is a sequence of random variables defined on a probability space \((\varOmega ,\mathscr {F}, P)\) and taking values in a complete separable metric space \(\mathscr {X}\). As is usual, we will refer to such a space as a Polish space . The metric of \(\mathscr {X}\) is denoted by d(xy), and expectation with respect to P by E. The theory of large deviations focuses on random variables \(\{X^{n}\}\) for which the probabilities \(P\{X^{n}\in A\}\) converge to 0 exponentially fast for a class of Borel sets A.

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Statistics and Operations ResearchUniversity of North CarolinaChapel HillUSA
  2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA

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