On Gaussian Measure of Balls in a Hilbert Space

Let $X$ be a centered Gaussian random vector taking values in a separable Hilbert space $H$, and let $a\in H$. We investigate the behavior of the density and the distribution function of a noncentered ball $\|X-a\|^2$ by means of its Laplace transform and obtain the results with an optimal estimate of the accuracy rate. As a tool we use a “local limit theorems” approach.