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Strong convergence of peaks over a threshold

Part of: Limit theorems Stochastic processes Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  23 August 2023

Simone A. Padoan  and
Stefano Rizzelli
Simone A. Padoan*
Affiliation:
Bocconi University
Stefano Rizzelli*
Affiliation:
Catholic University
*
*Postal address: Department of Decision Sciences, via Roentgen 1, 20136, Milan, Italy. Email: simone.padoan@unibocconi.it
**Postal address: Department of Statistical Science, via Lanzone 18, 20123, Milan, Italy. Email: stefano.rizzelli@unicatt.it
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Abstract

Extreme value theory plays an important role in providing approximation results for the extremes of a sequence of independent random variables when their distribution is unknown. An important one is given by the generalised Pareto distribution $H_\gamma(x)$ as an approximation of the distribution $F_t(s(t)x)$ of the excesses over a threshold t, where s(t) is a suitable norming function. We study the rate of convergence of $F_t(s(t)\cdot)$ to $H_\gamma$ in variational and Hellinger distances and translate it into that regarding the Kullback–Leibler divergence between the respective densities.

Keywords

Convergence rateexceedancesextreme quantilegeneralised Paretotail index

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60F15: Strong theorems 60B10: Convergence of probability measures
Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 2 , June 2024 , pp. 529 - 539
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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