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Strong convergence of peaks over a threshold
- Part of
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- Journal:
- Journal of Applied Probability / Volume 61 / Issue 2 / June 2024
- Published online by Cambridge University Press:
- 23 August 2023, pp. 529-539
- Print publication:
- June 2024
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- Article
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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.
