Upper estimate of overrunning by Sub_φ(Ω) random process the level specified by continuous function

Keywords: -sub-Gaussian stochastic process, metric entropy, supremum distribution, process of Ornstein-Uhlenbeck

In this paper we consider random process from the space Sub_φ(Ω), which is defined on compact set, and the probability that supremum of this process exceeds some function. The class of Sub_φ(Ω) random processes is more general than the class of Gaussian processes. By applying obtained estimation to a fluid queue fed by a process of Ornstein-Uhlenbeck from the space strictly Sub_φ(Ω), where

we show that for interval [a, b] there exist constants A, B, D and

for large enough buffer capacity x.