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

In the present paper, the random series ${\sum }_{m=0}^{\mathrm{\infty }}{c}_m{C}_m(\varpi )q_m^{(\zeta ,\eta )}(u)$ 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 $\zeta ,\eta \ge 0,$ 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_{ζ,η}(v,\varpi). \end{equation*} We also establish the existence of this integral in the sense of probability for $\mathfrak{g}\in {\mathrm{L}}_{[-1,1]}^{\mathrm{p}}(d{\mu }_{\zeta ,\eta })\mathrm{.}$