  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.
