Modeling of stochastic processes in $L_p(T)$ using orthogonal polynomials
Oleksandr Mokliachuk
TL;DR
The paper addresses modeling of second-order stochastic processes in $L_p(0,T)$ within the $Sub_\varphi(\Omega)$ framework, focusing on representations $X(t)=\sum_{k=0}^{\infty} \xi_k a_k(t)$ with independent, standardized coefficients $E\xi_k=0$, $E\xi_k\xi_l=\delta_{kl}$. It develops finite-dimensional models $X_N(t)=\sum_{k=0}^N \xi_k \hat{a}_k(t)$ using approximations $\hat{a}_k(t)$ when explicit $a_k(t)$ are unavailable, and provides reliability-accuracy bounds in $L_p(0,T)$ controlled by a computable quantity $c_N$. The work specializes to Hermite and Chebyshev polynomial bases, deriving explicit bounds via generating functions and auxiliary quantities such as $Z_f(t,\lambda)$ and $D_T(\omega)$, ensuring the approximations meet prescribed accuracy with probability at least $1-\alpha$. These results enable practical stochastic-process modeling in $L_p(0,T)$ even when explicit spectral decompositions are intractable, by leveraging orthogonal polynomial expansions and computable error controls.
Abstract
In this paper, models that approximate stochastic processes from the space $Sub_\varphi(Ω)$ with given reliability and accuracy in $L_p(T)$ are considered for some specific functions $\varphi(t)$. For processes that are decomposited in series using orthonormal bases, such models are constructed in the case where elements of such decomposition cannot be found explicitly.