## Random process from the class $$V(\varphi,\psi)$$: exceeding a curve.(English)Zbl 1164.60029

The authors consider random processes of the class $$V(\varphi,\psi)$$ defined on a compact set and the probability that this process exceeds some function. Basic definitions and some properties of $$\varphi$$-sub-Gaussian and strictly $$\varphi$$-sub-Gaussian spaces of random variables and processes are given. The authors obtain general results on estimates of probability that random process from the class $$V(\varphi,\psi)$$ overruns a level specified by a continuous function. The authors apply the obtained results to the generalized process of fractional Brownian motion of the class $$V(\varphi,\psi)$$ and obtain the estimate of overcrossing by its trajectories as the level defined by the function $$ct$$, where $$c>0$$ is a given constant. Such an estimate has applications in queuing theory as an estimate of buffer overflow probability or in risk theory as an estimate of ruin probability.

### MSC:

 60G20 Generalized stochastic processes 60G18 Self-similar stochastic processes 60K25 Queueing theory (aspects of probability theory)