Sojourn Times of Gaussian Processes with Trend

We derive exact tail asymptotics of sojourn time above the level $$u\ge 0$$ u ≥ 0 $$\begin{aligned} {\mathbb {P}} \left( v(u)\int _0^T {\mathbb {I}}(X(t)-ct>u)\text {d}t>x \right) , \quad x\ge 0, \end{aligned}$$ P v ( u ) ∫ 0 T I ( X ( t ) - c t > u ) d t > x , x ≥ 0 , as $$u\rightarrow \infty $$ u → ∞ , where X is a Gaussian process with continuous sample paths, c is some constant, v(u) is a positive function of u and $$T\in (0,\infty ]$$ T ∈ ( 0 , ∞ ] . Additionally, we analyze asymptotic distributional properties of $$\begin{aligned} \tau _u(x):=\inf \left\{ t\ge 0: {v(u)} \int _0^t {\mathbb {I}}(X(s)-cs>u)\text {d}s>x\right\} , \quad x \ge 0, \end{aligned}$$ τ u ( x ) : = inf t ≥ 0 : v ( u ) ∫ 0 t I ( X ( s ) - c s > u ) d s > x , x ≥ 0 , as $$u\rightarrow \infty $$ u → ∞ , where $$\inf \emptyset =\infty $$ 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.