Quasi-Banach spaces of random variables and stochastic processes

TL;DR

This book develops an advanced theory of quasi-Banach spaces of random variables and stochastic processes, extending classical spaces like Orlicz spaces, with applications in approximation, reliability, and Monte Carlo methods.

Contribution

It introduces new quasi-Banach $K_\sigma$-spaces of stochastic processes and explores their properties, distribution estimates, and modeling techniques, extending classical frameworks.

Findings

Distribution estimates for process suprema derived

Reliability and accuracy of stochastic models evaluated

Monte Carlo methods developed for multiple integrals

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.