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In the present paper, the random series $\sum\limits_{m=0}^\infty c_mC_m(\varpi)q_m^{(\zeta,\eta)}(u)$ in orthogonal Jacobi polynomials$q_m^{(\zeta,\eta)}(u)$ is discussed. The scalars $c_m$ are Fourier--Jacobicoefficients of a function in the weighted space$\mathrm{L}_{[-1,1]}^\mathrm{p}(d\mu_{\zeta,\eta}),\mathrm{p}>1.$ The randomvariables $C_m(\varpi)$ are chosen to be the Fourier--Jacobi coefficients ofsymmetric 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, undercertain conditions on $\mathrm{p},\zeta$ and $\eta,$ the random Fourier--Jacobiseries converges in probability to the stochastic integral \begin{equation*}\int_{-1}^1 \mathfrak{g}(u,v)dY_{\zeta,\eta}(v,\varpi). \end{equation*} We alsoestablish the existence of this integral in the sense of probability for$\mathfrak{g} \in \mathrm{L}_{[-1,1]}^\mathrm{p}(d\mu_{\zeta,\eta}).$

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