Piterbarg Theorems for Chi-processes with Trend

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 behavior of P{\sup_{t\in[0,T]} χ_n(t)>u} as u \to \infty, where $T$ is a given positive constant, and $g(\cdot )$ is some non-negative bounded measurable function. The case $g(t)\equiv 0$ is 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.

22 pages

References in corpus (1)

• On asymptotic constants in the theory of extremes for Gaussian processes