UNIVERSITY
OF WROCŁAW
 
Main Page
Contents
Online First
General Information
Instructions for authors


VOLUMES
43.1 42.2 42.1 41.2 41.1 40.2 40.1
39.2 39.1 38.2 38.1 37.2 37.1 36.2
36.1 35.2 35.1 34.2 34.1 33.2 33.1
32.2 32.1 31.2 31.1 30.2 30.1 29.2
29.1 28.2 28.1 27.2 27.1 26.2 26.1
25.2 25.1 24.2 24.1 23.2 23.1 22.2
22.1 21.2 21.1 20.2 20.1 19.2 19.1
18.2 18.1 17.2 17.1 16.2 16.1 15
14.2 14.1 13.2 13.1 12.2 12.1 11.2
11.1 10.2 10.1 9.2 9.1 8 7.2
7.1 6.2 6.1 5.2 5.1 4.2 4.1
3.2 3.1 2.2 2.1 1.2 1.1
 
 
WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 40, Fasc. 2,
pages 297 - 315
DOI: 10.37190/0208-4147.40.2.6
Published online 3.7.2020
 

Pickands--Piterbarg constants for self-similar Gaussian processes

Krzysztof Dębicki
Kamil Tabiś

Abstract: For a centered self-similar Gaussian process \(\{Y(t):t\in[0,\infty)\}\) and \(R\ge0\) we analyze the asymptotic behavior of \[\mathcal{H}_Y^R(T) = \mathbf{E} \exp \left( \sup_{t \in [0,T]} \bigl(\sqrt{2}\, Y(t) - (1+R) \sigma_Y^2(t) \bigr)\right)\] as \(T\to\infty\). We prove that \(\mathcal{H}_Y^R=\lim_{T\to\infty} \mathcal{H}_Y^R(T)\in(0,\infty)\) for \(R>0\) and \[\mathcal{H}_Y=\lim_{T\to\infty} \frac{\mathcal{H}_Y^0(T)}{T^\gamma}\in(0,\infty)\] for suitably chosen \(\gamma>0\). Additionally, we find bounds for \(\mathcal{H}_Y^R\), \(R>0\), and a surprising relation between \(\mathcal{H}_Y\) and the classical Pickands constants.

2010 AMS Mathematics Subject Classification: Primary 60G15; Secondary 60G70.

Keywords and phrases: Gaussian process, extremes, Pickands constant, Piterbarg constant

Download:    Abstract    Full text   Abstract + References