[Paper Review] Estimation of reliability and accuracy of models of $φ$-sub-Gaussian process using generating functions of polynomial expansions

Oleksandr Mokliachuk|arXiv (Cornell University)|Feb 4, 2026

Probabilistic and Robust Engineering Design0 citations

TL;DR

Extends reliability and accuracy estimates for 𝜙-sub-Gaussian processes to orthonormal polynomial systems without closed-form generating functions, including Legendre, generalized Laguerre, and Gegenbauer polynomials.

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 $φ$-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.

Motivation & Objective

• Motivate practical modeling of stochastic processes via orthonormal polynomial expansions when coefficient functions are not analytically available.
• Develop reliability and accuracy bounds that account for both truncation error and coefficient-approximation error.
• Extend previous 𝜙-sub-Gaussian framework to Legendre, generalized Laguerre, and Gegenbauer polynomial families.
• Provide actionable criteria for selecting the number of series terms to achieve prescribed reliability and accuracy.

Proposed method

• Represent stochastic processes as expansions with approximated coefficients X_N(t)=∑_{k=0}^{N} ξ_k â_k(t) and define the modeling error Δ_N(t)=X(t)−X_N(t).
• Use 𝜙-sub-Gaussian tails and the τ_𝜙 norm to bound the error dynamics via C_N=∫_0^T (τ_𝜙(Δ_N(t)))^p dμ(t).
• Derive polynomial-specific upper bounds for τ_𝜙(ξ_k) for Legendre, generalized Laguerre, and Gegenbauer bases.
• Obtain explicit C_N bounds in L_p([0,T]) and extend to C([0,T]) through additional regularity assumptions and generating-function considerations.
• Express resulting error bounds in terms of the generating-function-like constructs or their surrogate bounds for each polynomial family.
• Provide criteria to determine the required number of terms N to meet prescribed reliability and accuracy.

Experimental results

Research questions

• RQ1 How can reliability and accuracy bounds for 𝜙-sub-Gaussian process models be extended to orthonormal polynomial bases that lack normalized analytical generating functions?
• RQ2 How do Legendre, generalized Laguerre, and Gegenbauer polynomial families influence the bounds and practical selection of series terms?
• RQ3 How can coefficient-approximation error be incorporated alongside truncation error in a unified framework?
• RQ4 What concrete bounds on C_N can be derived for these polynomial bases in both L_p([0,T]) and C([0,T]) settings?
• RQ5 What practical criteria emerge for choosing N to meet specified reliability and accuracy levels?

Key findings

• New bounds are established for models in L_p([0,T]) and C([0,T]) that simultaneously account for truncation and coefficient-approximation errors.
• For Legendre polynomials, a bound for τ_𝜙(Δ_N(t)) is derived using a generating-function-based surrogate and leads to a C_N bound involving an explicit log-term expression.
• For generalized Laguerre polynomials, a bound on C_N involves Gamma functions and incomplete gamma-related expressions.
• For Gegenbauer polynomials, a bound on C_N incorporates hypergeometric and regularized hypergeometric functions.
• The results enable practical criteria to determine the number of series terms N required to achieve prescribed reliability and accuracy across the three polynomial families.
• The approach broadens applicability beyond Hermite and Chebyshev systems by accommodating polynomial families without closed-form generating functions.