On the Convergence of Random Fourier--Jacobi Series in weighted $\mathrm{L}_{[-1,1]}^{\mathrm{p},(ζ,η)}$ Space

Abstract: In the present paper, the random series $\sum\limits_{m=0}^\infty c_m C_m(\varpi)q_m^{(\zeta,\eta)}(u)$ in orthogonal Jacobi polynomials $q_m^{(\zeta,\eta)}(u)$ is discussed. The scalars $c_m$ are Fourier--Jacobi coefficients of a function in the weighted space $\mathrm{L}{[-1,1]}^\mathrm{p}(d\mu{\zeta,\eta}),\mathrm{p}>1.$ The random variables $C_m(\varpi)$ are chosen to be the Fourier--Jacobi coefficients of symmetric stable process $Y_{\zeta,\eta}(v,\varpi)$ of index $\chi \in [1,2]$ for $\zeta,\eta \geq 0,$ which are not independent. We prove that, under certain conditions on $\mathrm{p},\zeta$ and $\eta,$ the random Fourier--Jacobi series converges in probability to the stochastic integral \begin{equation*} \int_{-1}^1 \mathfrak{g}(u,v)dY_{\zeta,\eta}(v,\varpi). \end{equation*} We also establish the existence of this integral in the sense of probability for $\mathfrak{g} \in \mathrm{L}{[-1,1]}^\mathrm{p}(d\mu{\zeta,\eta}).$