Let $\textbf{ Z}(t)=(Z_1(t) ,\ldots, Z_d(t))^\top , t \in \mathbb{ R}$ where $Z_i(t), t\in \mathbb{ R}$, $i=1,...,d$ are mutually independent centered Gaussian processes with continuous sample paths a.s. and stationary increments. For $\textbf{ X}(t)= A \textbf{ Z}(t),\ t\in\mathbb{ R}$, where $A$ is a nonsingular $d\times d$ real-valued matrix, $\textbf{ u}, \textbf{ c}\in\mathbb{ R}^d$ and $T>0$ we derive tight bounds for \[ \mathbb{ P}\left\{ \exists_{ t\in [0,T]}: \cap_{ i=1}^d \{ X_i(t)- c_i t > u_i\}\right\} \] and find exact asymptotics as $(u_1,...,u_d)^{ \top}= (u a_1,..., ua_d)^\top$ and $u\to\infty$.
Categories: math.PR