Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Modelling with given reliability and accuracy in the space $L_p(T)$ of stochastic processes from $Sub_\varphi(Ω)$ decomposable in series with independent elements

Oleksandr Mokliachuk

TL;DR

The paper develops a general framework for modeling processes from $Sub_\varphi(\Omega)$ in $L_p(T)$ with prescribed reliability $1-\alpha$ and accuracy $\delta$, including representations as series of independent $\xi_k$ with deterministic coefficients. It derives probabilistic bounds using the Young-Fenchel transform $\varphi^*$ and Orlicz-type control $c_N$, and formulates concrete conditions to guarantee $P\{ \|X-X_N\|_{L_p(T)}>\delta\}\le \alpha$ for chosen $(\alpha,\delta)$. It also extends to processes whose expansion terms are not explicitly known by allowing approximations $\hat a_k$ and $\hat \lambda_k$, with error terms $\delta_k(t)$ and $\eta_k$, and provides Karhunen-Loève-based models when explicit eigenpairs are unavailable. As an application, the Karhunen-Loève decomposition is treated even when the integral equation cannot be solved in closed form, via approximated eigenfunctions/eigenvalues and Mercer's representation.

Abstract

Models that approximate stochastic processes from $Sub_\varphi(Ω)$ with given reliability and accuracy in $L_p(T)$ for some given $\varphi(t)$ are considered. We also study construction of models of processes which can be decomposed into series with approximate elements. Karhunen-Lo{è}ve model is considered as an example of the application of the proposed construction.

Modelling with given reliability and accuracy in the space $L_p(T)$ of stochastic processes from $Sub_\varphi(Ω)$ decomposable in series with independent elements

TL;DR

The paper develops a general framework for modeling processes from in with prescribed reliability and accuracy , including representations as series of independent with deterministic coefficients. It derives probabilistic bounds using the Young-Fenchel transform and Orlicz-type control , and formulates concrete conditions to guarantee for chosen . It also extends to processes whose expansion terms are not explicitly known by allowing approximations and , with error terms and , and provides Karhunen-Loève-based models when explicit eigenpairs are unavailable. As an application, the Karhunen-Loève decomposition is treated even when the integral equation cannot be solved in closed form, via approximated eigenfunctions/eigenvalues and Mercer's representation.

Abstract

Models that approximate stochastic processes from with given reliability and accuracy in for some given are considered. We also study construction of models of processes which can be decomposed into series with approximate elements. Karhunen-Lo{è}ve model is considered as an example of the application of the proposed construction.
Paper Structure (5 sections, 11 theorems, 87 equations)

This paper contains 5 sections, 11 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Estimation of reliability and accuracy of modelling of stochastic process in $L_p(T)$ spaces
  3. Construction models of stochastic processes from $Sub_\varphi(\Omega)$ that can be represented as a series with independent elements.
  4. Constructing models of stochastic processes in $L_p(0,T)$ using the Karhunen-Loève decomposition
  5. Conclusions

Key Result

Theorem 1

koz-kam Let $\{\mathbb T,\Lambda, M\}$ be a measurable space, let $X=\{X(t),t\in\mathbb T\} \subset S u b_\varphi (\Omega)$, and let $\tau_\varphi(t)=\tau_\varphi(X(t))$. Let the integral exist. Then the integral exists with probability 1 and for all where $f$ is a function such that $\varphi(u)=\int_0^u f(v)dv$$\forall u>0$ and where $\varphi^*$ is the Young-Fenchel transform of $\varphi$.

Theorems & Definitions (20)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • ...and 10 more