Exact Behavior of Gaussian Measures of Translated Balls in Hilbert Spaces

The distribution of a positive quadratic form of an infinite Gaussian sequence is investigated. Equivalently, the law of ||X + a||2 is described, where X is a symmetric Gaussian random variable taking values in a Hilbert space H and a ∈ H. It is shown that ||X + a||2 =d||X||2 + ξa, where ξa ≥ 0 is infinitely divisible and independent of ||X||2. Using this observation, various properties of the map a → P{ ||X + a||2 ≤ r}, r > 0, are derived. In particular, it is shown that this function is twice Gateaux differentiable at zero, the corresponding derivative is evaluated and a simple proof of Żak′s theorem (1989, Probab. Math. Statist. 10 257-270; Lecture Notes in Mathematics, Vol. 1391, pp. 401-405 Springer, New York/Berlin) is provided. Some applications for H = R2 are also discussed.