Quasi-Banach spaces of random variables and stochastic processes

Abstract

This book develops the theory of quasi-Banach $K_\sigma$-spaces $\mathbf{F}_\psi(\Omega)$, $\mathbf{F}_\psi^*(\Omega)$, and $D_{V,W}(\Omega)$ of random variables and stochastic processes, extending the classical framework of Orlicz spaces, $Sub_\varphi(\Omega)$ and $V(\varphi,\psi)$ 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 $\mathbf{F}_\psi(\Omega)$ are studied in detail. The third chapter derives distribution estimates for suprema of processes from $\mathbf{F}_\psi^*(\Omega)$, and the fourth addresses approximation theory in $SF_\psi(\Omega)$. The fifth chapter examines Orlicz spaces and their connections to $\mathbf{F}_\psi(\Omega)$. Chapters six and seven treat the pre-Banach $K_\sigma$-spaces $D_{V,W}(\Omega)$, 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 $\mathbf{F}_\psi(\Omega)$. 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 $Sub_\varphi(\Omega)$ 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.