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

Abstract

Stochastic processes are often represented through orthonormal series expansions, a framework originating in the classical works of Lo\`eve 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 $L_p(T)$ space 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.