The Metric of Large Deviation Convergence

Tiefeng Jiang; George L. O'Brien

doi:10.1023/A:1007814729591

The Metric of Large Deviation Convergence

Tiefeng Jiang
, George L. O'Brien

Statistics (Twin Cities)

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

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

Original language	English (US)
Pages (from-to)	805-824
Number of pages	20
Journal	Journal of Theoretical Probability
Volume	13
Issue number	3
DOIs	https://doi.org/10.1023/A:1007814729591
State	Published - 2000

Bibliographical note

Funding Information:
1Department of Statistics, 370 Serra Mall, Stanford University, Stanford, CA 94305-4065. E-mail: tjiang stat.stanford.edu. Present address: School of Statistics, 313 Ford Hall, 224 Church Street S.E., Minneapolis, MN 55455. 2Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada M2N 3T5. E-mail: obrien yorku.ca. Research of this author supported in part by the Natural Sciences and Engineering Research Council of Canada.

Keywords

Large deviations
Metric spaces

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10.1023/A:1007814729591

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title = "The Metric of Large Deviation Convergence",

abstract = "We construct a metric space of set functions (script Q sign(script X sign), d) such that a sequence \{Pn\} of Borel probability measures on a metric space (script X sign, d*) satisfies the full Large Deviation Principle (LDP) with speed \{an\} and good rate function I if and only if the sequence \{Pann\} converges in (script Q sign(script X sign), d) to the set function e-I. Weak convergence of probability measures is another special case of convergence in (script Q sign(script X sign), d). Properties related to the LDP and to weak convergence are then characterized in terms of (script Q sign(script X sign), d).",

keywords = "Large deviations, Metric spaces",

author = "Tiefeng Jiang and O'Brien, \{George L.\}",

note = "Funding Information: 1Department of Statistics, 370 Serra Mall, Stanford University, Stanford, CA 94305-4065. E-mail: tjiang stat.stanford.edu. Present address: School of Statistics, 313 Ford Hall, 224 Church Street S.E., Minneapolis, MN 55455. 2Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada M2N 3T5. E-mail: obrien yorku.ca. Research of this author supported in part by the Natural Sciences and Engineering Research Council of Canada.",

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AU - O'Brien, George L.

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PY - 2000

Y1 - 2000

N2 - We construct a metric space of set functions (script Q sign(script X sign), d) such that a sequence {Pn} of Borel probability measures on a metric space (script X sign, d*) satisfies the full Large Deviation Principle (LDP) with speed {an} and good rate function I if and only if the sequence {Pann} converges in (script Q sign(script X sign), d) to the set function e-I. Weak convergence of probability measures is another special case of convergence in (script Q sign(script X sign), d). Properties related to the LDP and to weak convergence are then characterized in terms of (script Q sign(script X sign), d).

AB - We construct a metric space of set functions (script Q sign(script X sign), d) such that a sequence {Pn} of Borel probability measures on a metric space (script X sign, d*) satisfies the full Large Deviation Principle (LDP) with speed {an} and good rate function I if and only if the sequence {Pann} converges in (script Q sign(script X sign), d) to the set function e-I. Weak convergence of probability measures is another special case of convergence in (script Q sign(script X sign), d). Properties related to the LDP and to weak convergence are then characterized in terms of (script Q sign(script X sign), d).

KW - Large deviations

KW - Metric spaces

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DO - 10.1023/A:1007814729591

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SN - 0894-9840

VL - 13

SP - 805

EP - 824

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

IS - 3

ER -

The Metric of Large Deviation Convergence

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