Piterbarg Theorems for Chi-processes with Trend

TL;DR

This paper establishes exact asymptotic results for the probability that a chi-process with trend exceeds a high threshold, extending Piterbarg's classical results to non-stationary Gaussian processes.

Contribution

It derives novel Piterbarg theorems for chi-processes with trend, applicable to both stationary and non-stationary Gaussian processes, expanding existing asymptotic theory.

Findings

Exact asymptotics for supremum probabilities of chi-processes with trend

Extension of Piterbarg theorems to non-stationary processes

Applicable to processes with a general trend function

Abstract

Let $\chi_n(t) = (\sum_{i=1}^n X_i^2(t))^{1/2},t\ge0$ 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]} \chi_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 $g(t)\equiv0$ 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.