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The Metric of Large Deviation Convergence

Author

Listed:
  • Tiefeng Jiang

    (Stanford University, Stanford)

  • George L. O'Brien

    (York University)

Abstract

We construct a metric space of set functions ( $$Q\left( X \right)$$ , d) such that a sequence {P n} of Borel probability measures on a metric space ( $$X$$ , d*) satisfies the full Large Deviation Principle (LDP) with speed {a n} and good rate function I if and only if the sequence $$\left\{ {P_n^{a_n } } \right\}$$ converges in ( $$Q\left( X \right)$$ , d) to the set function e −I . Weak convergence of probability measures is another special case of convergence in ( $$Q\left( X \right)$$ , d). Properties related to the LDP and to weak convergence are then characterized in terms of ( $$Q\left( X \right)$$ , d).

Suggested Citation

  • Tiefeng Jiang & George L. O'Brien, 2000. "The Metric of Large Deviation Convergence," Journal of Theoretical Probability, Springer, vol. 13(3), pages 805-824, July.
    • Handle: RePEc:spr:jotpro:v:13:y:2000:i:3:d:10.1023_a:1007814729591
    DOI: 10.1023/A:1007814729591
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    Keywords

    large deviations; metric spaces;

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