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 + a2 is described, where X is a symmetric Gaussian random variable taking values in a Hilbert space H and a [set membership, variant] H. It is shown that X + a2 =dX2 + [xi]a, where [xi]a >= 0 is infinitely divisible and independent of X2. Using this observation, various properties of the map a --> P{ X + a2 <= 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 Zak'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.