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A non-exponential extension of Sanov’s theorem via convex duality

Part of: Miscellaneous applications of functional analysis Limit theorems

Published online by Cambridge University Press:  29 April 2020

Daniel Lacker
Daniel Lacker*
Affiliation:
Columbia University
*
*Postal address: 306 Mudd, 500 West 120th St, New York, NY 10027, USA. Email address: [email protected]
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Abstract

This work is devoted to a vast extension of Sanov’s theorem, in Laplace principle form, based on alternatives to the classical convex dual pair of relative entropy and cumulant generating functional. The abstract results give rise to a number of probabilistic limit theorems and asymptotics. For instance, widely applicable non-exponential large deviation upper bounds are derived for empirical distributions and averages of independent and identically distributed samples under minimal integrability assumptions, notably accommodating heavy-tailed distributions. Other interesting manifestations of the abstract results include new results on the rate of convergence of empirical measures in Wasserstein distance, uniform large deviation bounds, and variational problems involving optimal transport costs, as well as an application to error estimates for approximate solutions of stochastic optimization problems. The proofs build on the Dupuis–Ellis weak convergence approach to large deviations as well as the duality theory for convex risk measures.

Keywords

Sanov’s theoremlarge deviationsconvex dualityrisk measuresweak convergenceempirical measuresheavy tailsstochastic optimization

MSC classification

Primary: 60F10: Large deviations
Secondary: 46N10: Applications in optimization, convex analysis, mathematical programming, economics
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 1 , March 2020 , pp. 61 - 101
Copyright
© Applied Probability Trust 2020

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