On Mean Convergence of Random Fourier - Hermite Series

The work in this article is an initiative to explore random Fourier - Hermite series in orthogonal Hermite polynomials. We choose the random coefficients in the series to be the Fourier-Hermite coefficients of a symmetric stable process with weight function , where . The existence of these random coefficients, which we find to be dependent random variables, is established. The random Fourier-Hermite series is proven to be convergent in the sense of mean if the scalars in the series are the Fourier-Hermite coefficients of a function  in the weighted space , where the weights are given by  with  such that . The sum functions of the series is obtained to the stochastic integral .