The Banach spaces \(\mathbf{F}_\psi (\Omega )\) of random variables (Q2922892)

This paper studies the space \(\mathbf F_\psi(\Omega)\) in connection with the theory of stochastic processes motivated by the original problem of obtaining accurate Monte Carlo approximations of integrals over a finite dimensional space depending on a parameter, a problem originally considered in [\textit{Yu. V. Kozachenko} and \textit{Yu. Yu. Mlavets}, Monte Carlo Methods Appl. 17, No. 2, 155--168 (2011; Zbl 1221.65011)]. The space considered consists of measurable functions \(\xi\) endowed with the norm NEWLINE\[NEWLINE \|{\xi}\|_\psi=\sup_{u\geq1}\frac{\operatorname{E}(|\xi|^u)^{1/u}}{\psi(u)},NEWLINE\]NEWLINE where \(\psi:\,]1,\infty[\to\mathbb R_+\) is increasing to \(\infty\) and continuous. The stochastic processes considered are collections indexed by a compact metric space of elements of some space as above. The authors establish some maximal as well as norm inequalities which follow from Theorem 4.1. The Monte Carlo applications are announced to be treated in a forthcoming sequel.