## 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 $$|c(t)|<1$$.

### MSC:

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