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Estimation of reliability and accuracy of models of $\varphi$-sub-Gaussian process using generating functions of polynomial expansions

Oleksandr Mokliachuk

TL;DR

This work broadens the reliability-accuracy analysis for phi-sub-Gaussian stochastic processes to orthonormal polynomial expansions that lack analytic generating functions, focusing on Legendre, generalized Laguerre, and Gegenbauer bases. By deriving computable bounds in $L_p(T)$ and $C([0,T])$ that jointly account for truncation and coefficient-approximation errors, the paper provides concrete criteria for the required number of terms in practical modeling. The results deliver explicit $C_N$-type bounds and gamma_N estimates, enabling practitioners to quantify and control modeling errors when coefficient expressions are unavailable. The framework expands the applicability of polynomial-based stochastic modeling and suggests avenues for extending to Jacobi polynomials and for numerical validation through future case studies.

Abstract

Stochastic processes are often represented through orthonormal series expansions, a framework originating in the classical works of Loève and Karhunen and widely used for simulation and numerical approximation. While truncation error in such expansions has been extensively studied, practical models frequently involve an additional source of error arising from the approximation of coefficient functions when closed-form expressions are unavailable. The combined effect of these two errors remains insufficiently addressed in the literature. Building on the author's earlier work on reliability and accuracy estimates for $\varphi$-sub-Gaussian processes, this paper extends the methodology to orthonormal polynomial systems that do not possess normalized generating functions in analytical form, including the Legendre, generalized Laguerre, and Gegenbauer families. New bounds are derived for models in $L_p(T)$ and $C([0,T])$ that simultaneously account for truncation and coefficient approximation. The resulting criteria provide practical guidance for selecting the number of series terms required to achieve prescribed levels of reliability and accuracy across a broader class of polynomial-based stochastic process models.

Estimation of reliability and accuracy of models of $\varphi$-sub-Gaussian process using generating functions of polynomial expansions

TL;DR

This work broadens the reliability-accuracy analysis for phi-sub-Gaussian stochastic processes to orthonormal polynomial expansions that lack analytic generating functions, focusing on Legendre, generalized Laguerre, and Gegenbauer bases. By deriving computable bounds in and that jointly account for truncation and coefficient-approximation errors, the paper provides concrete criteria for the required number of terms in practical modeling. The results deliver explicit -type bounds and gamma_N estimates, enabling practitioners to quantify and control modeling errors when coefficient expressions are unavailable. The framework expands the applicability of polynomial-based stochastic modeling and suggests avenues for extending to Jacobi polynomials and for numerical validation through future case studies.

Abstract

Stochastic processes are often represented through orthonormal series expansions, a framework originating in the classical works of Loève and Karhunen and widely used for simulation and numerical approximation. While truncation error in such expansions has been extensively studied, practical models frequently involve an additional source of error arising from the approximation of coefficient functions when closed-form expressions are unavailable. The combined effect of these two errors remains insufficiently addressed in the literature. Building on the author's earlier work on reliability and accuracy estimates for -sub-Gaussian processes, this paper extends the methodology to orthonormal polynomial systems that do not possess normalized generating functions in analytical form, including the Legendre, generalized Laguerre, and Gegenbauer families. New bounds are derived for models in and that simultaneously account for truncation and coefficient approximation. The resulting criteria provide practical guidance for selecting the number of series terms required to achieve prescribed levels of reliability and accuracy across a broader class of polynomial-based stochastic process models.
Paper Structure (9 sections, 5 theorems, 121 equations)

This paper contains 9 sections, 5 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Basic notations
  3. Model estimations in $L_p([0,T])$
  4. Legendre polynomials
  5. Generalized Laguerre polynomials
  6. Gegenbauer polynomials
  7. Model estimations in $C([0,T])$
  8. Discussion and Conclusions
  9. Declarations

Key Result

Theorem 2.1

kozroztur (On decomposition of the stochastic process using an orthonormal basis) Assume $X(t),t\in \mathbf{T}$ is a stochastic process of the second order, $EX(t) = 0$$\forall t \in \mathbf{T}$. Let $B(t, s) = EX(t)\overline{X(s)}$ be the autocorrelation function of $X(t)$. Assume $f(t, \lambda)$ i if and only if where Here, $\xi_k$ are centered uncorrelated random variables: $E\xi_k=0$, $E\xi_

Theorems & Definitions (13)

  • Theorem 2.1
  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Theorem 3.1
  • Theorem 3.2
  • ...and 3 more