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Liu and Liu introduced the random Fourier transform, which is a randomFourier series in Hermite functions, and applied it to image encryption anddecryption. They expected its applications in optics and informationtechnology. These motivated us to look into random Fourier series in orthogonalpolynomials. Recently, we have established the convergence of randomFourier--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 withstochastic processes like the Wiener process, the symmetric stable process, andthe scalars $d_n$ are the Fourier--Jacobi coefficients of functions in someclasses of continuous functions. It is observed that the mode of convergence ofthe random series depends on the choice of the scalars $d_n$ and the stochasticprocesses. In this article, we have investigated the scalars $d_n,$ which arechosen 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.

On the Convergence of Random Fourier--Jacobi Series in $L_{[-1,1]}^{p,(\eta,\tau)}$ space