Asymptotics of supremum distribution of a Gaussian process over a Weibullian time

Let $\{X(t):t\in [0,\infty)\}$ be a centered Gaussian process with stationary increments and variance function $\sigma^2_X(t)$. The classical result of Berman shows that under some %regularity conditions on the variance function $\sigma^2_X(\cdot)$, providing it is convex, the following asymptotics holds for deterministic $T$: \[ \mathbb{P}(\sup_{t\in[0,T]}X(t)>u)=\mathbb{P}(X(T) > u)(1+o(1)), \] as $u\to\infty$. We extend this result to the case of random $T$ being asymptotically Weibullian, that is \[ \mathbb{P}(T>t)= Ct^{\gamma}\exp(-\beta t^\alpha) (1 + o(1)), \] as $t \to \infty$, where $\alpha, \beta, C > 0, \gamma \in \mathbb{R}$. We apply obtained asymptotics to analyze of extremal behaviour of fractional Laplace motion process.