Stochastic processes of the classes \(V(\varphi,\psi)\). (English. Ukrainian original) Zbl 0990.60036

Theory Probab. Math. Stat. 63, 109-121 (2001); translation from Teor. Jmovirn. Mat. Stat. 63, 100-111 (2000).
Let us denote by \(\text{Sub}_{\varphi}(\Omega)\) the space of centered random variables \(\xi\) such that for all \(\lambda\in R\) there exists a constant \(r_{\xi}\geq 0\) such that \(E\exp\{\lambda\xi\}\leq\exp\{\varphi(\lambda r_{\xi})\}\). Let \((T,\rho)\) be a pseudo-metric space. A random process \(x(t), t\in T\), belongs to the class \(V(\varphi,\psi)\), where \(\varphi\prec\psi\), if \(x(t)\in \text{Sub}_{\psi}(\Omega)\) for all \(t\in T\) and \(x(t)-x(s)\in \text{Sub}_{\varphi}(\Omega)\) for all \(s,t\in T\). Here \(\varphi, \psi\) are the Orlicz \(N\)-functions. The authors obtain estimates for \(P\{\sup_{t\in T}|c(t)x(t)|\geq\varepsilon\}\), where \(x(t)\) is a separable random process from the class \(V(\varphi,\psi)\), \(c(t)\) is a continuous function such that \(|c(t)|<1\).


60G07 General theory of stochastic processes
60G70 Extreme value theory; extremal stochastic processes