Extremes

, Volume 18, Issue 1, pp 37–64 | Cite as

Piterbarg theorems for chi-processes with trend

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Abstract

Let \(\chi _{n}(t) = ({\sum }_{i=1}^{n} {X_{i}^{2}}(t))^{1/2},\ {t\ge 0}\) 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
$$ \mathbb{P}\left\{\sup\limits_{t\in[0,T]} \left(\chi_{n}(t)- {g(t)} \right) > u\right\} \;\; \text{as} \;\; u \rightarrow \infty, $$
(1)
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.

Keywords

Gaussian random fields Piterbarg theorem for chi-process Pickands constant generalized Piterbarg constant Piterbarg inequality 

AMS 2000 Subject Classifications

Primary–60G15 Secondary–60G70 
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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Faculty of Business and Economics (HEC Lausanne)University of LausanneUNIL-DorignySwitzerland

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