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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 genericcentered Gaussian process $X$. This paper derives the exact asymptotic behaviorof P{\sup_{t\in[0,T]} \chi_n(t)>u} as u \to \infty, where $T$ is a givenpositive constant, and $g(\cdot)$ is some non-negative bounded measurablefunction. The case $g(t)\equiv0$ is investigated in numerous contributions byV.I. Piterbarg. Our novel asymptotic results for both stationary andnon-stationary $X$are referred to as Piterbarg theorems for chi-processes withtrend.

Piterbarg Theorems for Chi-processes with Trend