Let \(\mu\) be a centered Gaussian measure on a separable Hilbert space \((E,||\cdot||)\), \(X\) be a \(\mu\)-distributed random variable in \(E\), denote by \(\{\lambda_j\}\) the non-increasing sequence of eigenvalues of the covariance operator of \(\mu\) and let \(\{e_j\}\) be a corresponding basis of orthonormal eigenfunctions. Consider the random variable \(-\log\mu(B(X,t))\), where \(B(x,r)\) denotes the closed ball in \(E\) with center \(x\in E\) and radius \(r\geq 0\). One of the main results of the paper is the following
Theorem 2.1. We have \[ \lim_{t\to{0+}}\{-\log\mu(B(X,t))/\Lambda^*(t^2)\}=1\text{ a.s.,} \] where \(\Lambda^*(t)=\sup_{-\infty<\theta<+\infty} [t\theta-\Lambda(\theta)]\) is the Legendre transform of \(\Lambda(\theta)=\sum_j[-{1\over 2}\log(1-2\theta\lambda_j)+ {\theta\lambda_j\over{1-2\theta\lambda_j}}]\), \(\theta<1/(2\lambda_1)\), and \(\Lambda(\theta)=\infty\) for \(\theta\in[1/2\lambda_1;+\infty]\).
Moreover it is proved that the asymptotic behavior of \(-\log\mu(B(X,t))\) is a.s. equivalent to that of a deterministic function \(\varphi_R(\varepsilon)\). These new insights are used to derive the precise asymptotics of a random quantization problem, which was introduced by the author, F. Fehringer, A. Matoussi and M. Scheutzow [J. Theor. Probab. 16, 249-265 (2003; Zbl 1017.60012)].