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Quasi-Banach spaces of random variables and stochastic processes

Yuriy Kozachenko, Yuriy Mlavets, Oleksandr Mokliachuk

Abstract

This book develops the theory of quasi-Banach $K_σ$-spaces $\mathbf{F}_ψ(Ω)$, $\mathbf{F}_ψ^*(Ω)$, and $D_{V,W}(Ω)$ of random variables and stochastic processes, extending the classical framework of Orlicz spaces, $Sub_\varphi(Ω)$ and $V(\varphi,ψ)$ spaces. The book consists of eleven chapters. The first two chapters establish the foundational theory: stochastic processes from quasi-Banach $K_σ$-spaces are introduced, and the fundamental properties of $\mathbf{F}_ψ(Ω)$ are studied in detail. The third chapter derives distribution estimates for suprema of processes from $\mathbf{F}_ψ^*(Ω)$, and the fourth addresses approximation theory in $SF_ψ(Ω)$. The fifth chapter examines Orlicz spaces and their connections to $\mathbf{F}_ψ(Ω)$. Chapters six and seven treat the pre-Banach $K_σ$-spaces $D_{V,W}(Ω)$, 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 $\mathbf{F}_ψ(Ω)$. 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 $Sub_\varphi(Ω)$ processes - subclasses of $K_σ$-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.

Quasi-Banach spaces of random variables and stochastic processes

Abstract

This book develops the theory of quasi-Banach -spaces , , and of random variables and stochastic processes, extending the classical framework of Orlicz spaces, and spaces. The book consists of eleven chapters. The first two chapters establish the foundational theory: stochastic processes from quasi-Banach -spaces are introduced, and the fundamental properties of are studied in detail. The third chapter derives distribution estimates for suprema of processes from , and the fourth addresses approximation theory in . The fifth chapter examines Orlicz spaces and their connections to . Chapters six and seven treat the pre-Banach -spaces , establishing their essential properties and evaluating reliability and accuracy of stochastic process models. The eighth chapter provides norm distribution estimates in for processes from . The ninth chapter develops the Monte Carlo method for multiple integrals over with prescribed reliability and accuracy. The final two chapters treat modeling of processes - subclasses of -spaces - with given reliability and accuracy in and respectively. The results are substantially based on the authors' original work and that of their co-authors.

Paper Structure

This paper contains 60 sections, 157 theorems, 1421 equations.

Table of Contents

  1. Random processes from quasi-Banach $K_{\sigma}$-spaces of random variables
  2. Quasi-Banach $K_{\sigma}$-spaces of random variables
  3. Orlicz spaces of random variables
  4. Spaces $M(\Omega)$ of random variables
  5. Random processes from quasi-Banach $K_{\sigma}$-spaces
  6. Random processes from quasi-Banach $K_{\sigma}$-spaces in the case where $T=\mathbb R^{1}$
  7. Spaces $\mathbf{F}_\psi(\Omega)$ of random variables
  8. Spaces $\mathbf{F}_\psi(\Omega)$ of random variables. Basic properties
  9. Spaces $\mathbf{F}_{S_k,\psi,r}(\Omega)$ of random variables
  10. Condition $\mathbf{H}$ for spaces $\mathbf{F}_\psi(\Omega)$
  11. Random processes from spaces $\mathbf{F}_\psi^*(\Omega)$
  12. Estimates for the distribution of suprema of random processes from spaces $\mathbf{F}_\psi^*(\Omega)$
  13. Probabilities of large deviations for sums of independent random processes from spaces $\mathbf{F}_\psi^*(\Omega)$
  14. Distribution of suprema of increments of random processes from spaces $\mathbf{F}_\psi(\Omega)$
  15. Conditions for convergence of wavelet decompositions of random processes from spaces $\mathbf{F}_\psi(\Omega)$
  16. ...and 45 more sections

Key Result

Lemma 1.1

Let $\xi_{n},n=1,2,...$ be a sequence of random variables such that $\xi_{n}\in K(\Omega)$, where $K(\Omega)$ is a quasi-$K_{\sigma}$-space. If there exists a random variable $\xi_{n}\in K(\Omega)$ such that $||\xi_{n}-\xi||\rightarrow0$ as $n\rightarrow\infty$, then $\xi_{n}\rightarrow\xi$ by proba

Theorems & Definitions (412)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Remark 1.1
  • Definition 1.6
  • Remark 1.2
  • Definition 1.7
  • Definition 1.8
  • ...and 402 more