Conditional limit theorems for queues with Gaussian input, a weak convergence approach

We consider a buffered queueing system that is fed by a Gaussian source and drained at a constant rate. The fluid offered to the system in a time interval $(0,t]$ is given by a separable continuous Gaussian process $Y$ with stationary increments. The variance function $sigma^2:tmapsto var Y_t$ of $Y$ is assumed to be regularly varying with index $2H$, for some $0