An asymptotic expansion of the distribution of a homogeneous functional of a strictly stable vector

The classical method of obtaining asymptotic expansions for stable distributions in the one-dimensional case is connected with contour integration. We use another approach based on the representation of a stable random variable in the form of a stochastic integral by Poisson random measure and on the decomposition of the distribution of this random measure into the sum of linear functionals adapted to the isolation of the separate terms of the asymptotics.