Extremes of the time-average of stationary Gaussian processes

We study the exact asymptotics of P ( sup t ≥ 0 I Z ( t ) > u ) , as u → ∞ , where I Z ( t ) = { 1 t ∫ 0 t Z ( s ) d s for t > 0 Z ( 0 ) for t = 0 and { Z ( t ) : t ≥ 0 } is a centered stationary Gaussian process with covariance function satisfying some regularity conditions. application, we analyze the probability of buffer emptiness in a Gaussian fluid queueing system and the collision probability of differentiable Gaussian processes with stationary increments. Additionally, we find estimates for analogues of Piterbarg–Prisyazhnyuk constants, that appear in the form of the considered asymptotics.