On Summability of Random Fourier-Jacobi Series associated with Stable Process

TL;DR

This paper investigates the convergence properties of random Fourier-Jacobi series associated with symmetric stable processes, establishing conditions under which they converge in mean and are summable in probability, depending on the stability index and function space.

Contribution

It extends the theory of Fourier-Jacobi series to stochastic settings involving stable processes, providing new convergence and summability results for these random series.

Findings

Series converges in mean to the stochastic integral for b1  (1,2]

Series is weakly continuous in probability under certain conditions

Series is (C,1) summable in probability when b1=1 and f is in a weighted continuous space

Abstract

Let $X(t,\omega),$ $t \in \textit{R}$ be a symmetric stable process with index $\alpha \in (1,2]$ and $a_n$ be the Fourier-Jacobi coefficients of $f \in L^p,$ where $p \geq \alpha.$ For $\gamma, \delta> 0,$ $t \in [-1,1],$ define $A_n(\omega)=\int_{-1}^1 P_n^{(\gamma,\delta)}(t)\rho^{(\gamma,\delta)}dX(t,\omega)$ where $P_n^{(\gamma,\delta)}(t)$ are orthogonal Jacobi polynomials. The $A_n(\omega)$ exists in the sense of mean. In this paper, it is shown that the random Fourier-Jacobi series $\sum_{n=0}^\infty a_n A_n(\omega)P_n^{(\gamma,\delta)}(y)$ converges to the stochastic integral $\int_{-1}^1f(y,t)\rho^{(\gamma,\delta)}dX(t,\omega)$ in the sense of mean and the sum function is weakly continuous in probability if the index $\alpha \in (1,2]$ and $f \in L^p$ where $P \geq \alpha.$ However, it is shown that if the index $\alpha$ is one and $f$ is in the weighted space of continuous function $C^{(\eta, \tau)}(-1,1),$ for $\eta, \tau \geq 0,$ then the random Fourier-Jacobi series is $(C,1)$ summable in probability to the stochastic integral $\int_{-1}^1f(y, t)\rho^{(\gamma,\delta)}dX(t,\omega).$