Asymptotical properties of distributions of isotropic Lévy processes

The asymptotic expressions are given in terms of the radial part of characteristic exponent ψ and its derivative. In particular, when ψ(λ)−λ2ψ′(λ) varies regularly, as tψ(r−1)2ψ(r−1)−(2r)−1ψ′(r−1)→0 the tail probability -(|Xt|≥r) is asymptotically equal to a constant times t(ψ(r−1)−(2r)−1ψ′(r−1)).