  In the present paper, we study the random series $\sum_{m=0}^\infty c_m
C_m(\varpi)q_m^{(\zeta,\eta)}(u)$ in orthogonal Jacobi polynomials
$q_m^{(\zeta,\eta)}(u),$ where the scalars $c_m$ are Fourier--Jacobi
coefficients of a function in the weighted space
$\mathrm{L}_{[-1,1]}^\mathrm{p}(d\mu_{\zeta,\eta}),\mathrm{p}>1$ with weighted
measure $d\mu_{\zeta,\eta}(u):=\varrho^{(\zeta,\eta)}(u)du,$ where
$\varrho^{(\zeta,\eta)}(u)$ is the Jacobi weight. The random variables
$C_m(\varpi)$ are chosen to be the Fourier--Jacobi coefficients
$Y_{\zeta,\eta}(v,\varpi),$ for $\zeta,\eta \geq 0$ which are not independent.
The $Y_{\zeta,\eta}(v,\varpi)$ defined as $\int_{a}^{v}
\varrho^{(\zeta,\eta)}(s)d\mathfrak{X}(s,\varpi)$ is found to be the symmetric
stable process with index same as that of $\mathfrak{X}(v,\varpi)$ of index
$\chi \in [1,2]$ and $a \in [-1,1].$ We prove that the random Fourier--Jacobi
series under some conditions on $\mathrm{p},\zeta$ and $\eta$ converges in
probability to the stochastic integral $\int_a^b
\mathfrak{g}(u,v)dY_{\zeta,\eta}(v,\varpi)$ in $[a,b],$ where $[a,b]$ lies
inside the orthogonality segment $[-1,1]$ in which the weight function
$\varrho^{(\zeta,\eta)}(v)$ is bounded by one. The existence of this integral
is also established, for $\mathfrak{g} \in
\mathrm{L}_{[-1,1]}^\mathrm{p}(d\mu_{\zeta,\eta}).$
