Large deviations built on max-stability

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2021
Authors
Zapata, José Miguel
Journal article
Published
Published in
Bernoulli ; 27 (2021), 2. - pp. 1001-1027. - International Statistical Institute. - ISSN 1350-7265
Abstract
In this paper, we show that the basic results in large deviations theory hold for general monetary risk measures, which satisfy the crucial property of max-stability. A max-stable monetary risk measure fulfills a lattice homomorphism property, and satisfies under a suitable tightness condition the Laplace Principle (LP), that is, admits a dual representation with affine convex conjugate. By replacing asymptotic concentration of probability by concentration of risk, we formulate a Large Deviation Principle (LDP) for max-stable monetary risk measures, and show its equivalence to the LP. In particular, the special case of the asymptotic entropic risk measure corresponds to the classical Varadhan–Bryc equivalence between the LDP and LP. The main results are illustrated by the asymptotic shortfall risk measure.
510 Mathematics
Cite This
ISO 690KUPPER, Michael, José Miguel ZAPATA, 2021. Large deviations built on max-stability. In: Bernoulli. International Statistical Institute. 27(2), pp. 1001-1027. ISSN 1350-7265. Available under: doi: 10.3150/20-BEJ1263
BibTex
@article{Kupper2021Large-53435,
year={2021},
doi={10.3150/20-BEJ1263},
title={Large deviations built on max-stability},
number={2},
volume={27},
issn={1350-7265},
journal={Bernoulli},
pages={1001--1027},
author={Kupper, Michael and Zapata, José Miguel}
}

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<dcterms:abstract xml:lang="eng">In this paper, we show that the basic results in large deviations theory hold for general monetary risk measures, which satisfy the crucial property of max-stability. A max-stable monetary risk measure fulfills a lattice homomorphism property, and satisfies under a suitable tightness condition the Laplace Principle (LP), that is, admits a dual representation with affine convex conjugate. By replacing asymptotic concentration of probability by concentration of risk, we formulate a Large Deviation Principle (LDP) for max-stable monetary risk measures, and show its equivalence to the LP. In particular, the special case of the asymptotic entropic risk measure corresponds to the classical Varadhan–Bryc equivalence between the LDP and LP. The main results are illustrated by the asymptotic shortfall risk measure.</dcterms:abstract>
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Yes
Yes