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Piterbarg theorems for chi-processes with trend

Hashorva, Enkelejd ; Ji, Lanpeng

In: Extremes, 2015, vol. 18, no. 1, p. 37-64

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    Titre
    • Piterbarg theorems for chi-processes with trend
    Auteur
    • Hashorva, Enkelejd. Faculty of Business and Economics (HEC Lausanne), University of Lausanne, 1015, UNIL-Dorigny, Lausanne, Switzerland
    • Ji, Lanpeng. Faculty of Business and Economics (HEC Lausanne), University of Lausanne, 1015, UNIL-Dorigny, Lausanne, Switzerland
    Type de document
    • Postprint
    Langue
    • Anglais
    Publié dans
    • Extremes, 2015, vol. 18, no. 1, p. 37-64. Springer US; http://www.springer-ny.com
    Autre version électronique
    Identifiant OAI-PMH
    • oai:doc.rero.ch:331638
    Summary
    Let χ n ( t ) = ( ∑ i = 1 n X i 2 ( t ) ) 1 / 2 , t ≥ 0 $\chi _{n}(t) = ({\sum }_{i=1}^{n} {X_{i}^{2}}(t))^{1/2},\ {t\ge 0}$ be a chi-process with n degrees of freedom where X i 's are independent copies of some generic centered Gaussian process X. This paper derives the exact asymptotic behaviour of 1 ℙ sup t ∈ [ 0 , T ] χ n ( t ) − g ( t ) > u as u → ∞ , $$ \mathbb{P}\left\{\sup\limits_{t\in[0,T]} \left(\chi_{n}(t)- {g(t)} \right) > u\right\} \;\; \text{as} \;\; u \rightarrow \infty, $$ where T is a given positive constant, and g(⋅) is some non-negative bounded measurable function. The case g(t)≡0 has been investigated in numerous contributions by V.I. Piterbarg. Our novel asymptotic results, for both stationary and non-stationary X, are referred to as Piterbarg theorems for chi-processes with trend.