  Liu and Liu introduced the random Fourier transform, which is a random
Fourier series in Hermite functions, and applied it to image encryption and
decryption. They expected its applications in optics and information
technology. These motivated us to look into random Fourier series in orthogonal
polynomials. Recently, we have established the convergence of random
Fourier--Jacobi series $\sum\limits_{n=0}^\infty d_n r_n(\omega)\varphi_n(y),$
where $\varphi_n(y)$ are the orthonormal Jacobi polynomials
$p_n^{(\gamma,\delta)}(y),$ $r_n(\omega)$ are random variables associated with
stochastic processes like the Wiener process, the symmetric stable process, and
the scalars $d_n$ are the Fourier--Jacobi coefficients of functions in some
classes of continuous functions. It is observed that the mode of convergence of
the random series depends on the choice of the scalars $d_n$ and the stochastic
processes. In this article, we have investigated the scalars $d_n,$ which are
chosen to be the Fourier--Jacobi coefficients of functions in some weighted
$L_{[-1,1]}^{p,(\eta,\tau )}$ spaces, so that the random series converges.
Further, the continuity property of the sum functions is studied.
