On the growth of Random Fourier-Hermite series

Abstract

The exact growth of the random Fourier–Hermite series Σ∞????=0????????????????(????)????????(????), and its Fourier transform are established. Here ????????(????) are considered to be both orthogonal Hermite functions in ????2(ℝ) and transformed Hermite function in ????2[0,1]. ???????? are the Fourier–Hermite coefficients of functions in different ????2 spaces. ????????(????) are random Fourier–Hermite coefficient associated with Wiener process and symmetric stable process of index 2.