Quasi-Banach spaces of random variables and stochastic processes

This book develops the theory of quasi-Banach $K_{\sigma }$-spaces ${\mathbf{F}}_{\psi }(\mathrm{\Omega })$, ${\mathbf{F}}_{\psi }^{\ast }(\mathrm{\Omega })$, and $D_{V,W}(\mathrm{\Omega })$ of random variables and stochastic processes, extending the classical framework of Orlicz spaces, $Su{b}_{\phi }(\mathrm{\Omega })$ and $V(\phi ,\psi )$ spaces. The book consists of eleven chapters. The first two chapters establish the foundational theory: stochastic processes from quasi-Banach $K_{\sigma }$-spaces are introduced, and the fundamental properties of ${\mathbf{F}}_{\psi }(\mathrm{\Omega })$ are studied in detail. The third chapter derives distribution estimates for suprema of processes from ${\mathbf{F}}_{\psi }^{\ast }(\mathrm{\Omega })$, and the fourth addresses approximation theory in $S{F}_{\psi }(\mathrm{\Omega })$. The fifth chapter examines Orlicz spaces and their connections to ${\mathbf{F}}_{\psi }(\mathrm{\Omega })$. Chapters six and seven treat the pre-Banach $K_{\sigma }$-spaces $D_{V,W}(\mathrm{\Omega })$, establishing their essential properties and evaluating reliability and accuracy of stochastic process models. The eighth chapter provides norm distribution estimates in $L_p(T)$ for processes from ${\mathbf{F}}_{\psi }(\mathrm{\Omega })$. The ninth chapter develops the Monte Carlo method for multiple integrals over ${\mathbb{R}}^n$ with prescribed reliability and accuracy. The final two chapters treat modeling of $Su{b}_{\phi }(\mathrm{\Omega })$ processes - subclasses of $K_{\sigma }$-spaces - with given reliability and accuracy in $L_p(T)$ and $C(T)$ respectively. The results are substantially based on the authors’ original work and that of their co-authors.