  We give a convergence proof for the approximation by sparse collocation of
Hilbert-space-valued functions depending on countably many Gaussian random
variables. Such functions appear as solutions of elliptic PDEs with lognormal
diffusion coefficients. We outline a general $L^2$-convergence theory based on
previous work by Bachmayr et al. (2016) and Chen (2016) and establish an
algebraic convergence rate for sufficiently smooth functions assuming a mild
growth bound for the univariate hierarchical surpluses of the interpolation
scheme applied to Hermite polynomials. We verify specifically for Gauss-Hermite
nodes that this assumption holds and also show algebraic convergence w.r.t. the
resulting number of sparse grid points for this case. Numerical experiments
illustrate the dimension-independent convergence rate.
