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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.

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  1. Faculty of Business and Economics (HEC Lausanne), University of Lausanne, UNIL-Dorigny, 1015, Lausanne, Switzerland

    Enkelejd Hashorva & Lanpeng Ji

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  1. Enkelejd Hashorva
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  2. Lanpeng Ji
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Correspondence to Lanpeng Ji.

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Hashorva, E., Ji, L. Piterbarg theorems for chi-processes with trend. Extremes 18, 37–64 (2015). https://doi.org/10.1007/s10687-014-0201-1

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