Distribution of the norm of a stable random vector of a Hilbert space (Q1085867)

Let B be a separable Banach space, X be a B-valued random variable with a stable distribution of exponent \(0<\alpha <2\) and f be a second order polynomial on B. The authors characterize the bounded continuous differentiability of \(F(x)={\mathbb{P}}(f(X)<x)\) in terms of the linear support \(B_ 1\) of \(\tilde X\) (the symmetrization of X), and the kernel space of A, the linear operator of \(B_ 1\) into \(B^*_ 1\) induced from the quadratic part of f. In particular, they prove that the distribution function F has bounded infinitely differentiable density if B is a Hilbert space, \(f(x)=| x|\) and the stable random variable X, \(1<\alpha <2\), is not concentrated in a finite dimensional subspace of B.