Sojourn Times of Gaussian Processes with Trend

We derive exact tail asymptotics of sojourn time above the level u≥ 0 P(v(u)∫0TI(X(t)-ct>u)dt>x),x≥0,as u→ ∞, where X is a Gaussian process with continuous sample paths, c is some constant, v(u) is a positive function of u and T∈ (0 , ∞]. Additionally, we analyze asymptotic distributional properties of τu(x):=inf{t≥0:v(u)∫0tI(X(s)-cs>u)ds>x},x≥0,as u→ ∞, where inf ∅ = ∞. The findings of this contribution are illustrated by a detailed analysis of a class of Gaussian processes with stationary increments and a family of self-similar Gaussian processes.