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.


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