Note relating Bochner integrals and reproducing kernels to series expansions on a Gaussian Banach space

Fernique’s recent proof of finiteness of positive moments of the norm of a Banach-valued Gaussian random vector $\mathfrak {X}$ is used to prove rth mean convergence of reproducing kernel series representations of $\mathfrak {X}$. Embedding of the reproducing kernel Hilbert space into the Banach range of X is explicitly given by Bochner integration. This work extends and clarifies work of Kuelbs, Jain and Kallianpur.