The Gaussian measure of shifted balls

SummaryLet μ be a centered Gaussian measure on a Hilbert spaceH and let $$B_R \subseteq H$$ be the centered ball of radiusR>0. Fora∈H and $$\mathop {\lim }\limits_{t{\mathbf{ }} \to {\mathbf{ }}\infty } {\mathbf{ }}R(t)/t< {\mathbf{ }}||a||$$ , we give the exact asymptotics of μ(BR(t)+t·a) ast→∞. Also, upper and lower bounds are given when μ is defined on an arbitrary separable Banach space. Our results range from small deviation estimates to large deviation estimates.