A new proof of the zero-one law for stable measures

PROOF OF THE THEOREM. We may assume that yt is strictly stable. (Indeed, if 0 0 such that 1ii(x)l < 1 for lxl < a where Ai is the Fourier transform of yt. Lettingfa(x) a 1(I-a x 1), -a < x < a, and zero elsewhere, we have