Asymptotic expansion of the distribution of a homogeneous functional of a strictly stable random vector. II

Let $u$ be a strictly stable non-Gaussian vector with the exponent of stability $\alpha\ge 1$, taking on values in a separable Banach space $B$. Let $h\colon B\to\mathbb R$ be a smooth homogeneous functional and let $F$ be the distribution function of the random variable $h(u)$. For the function $1-F(x)$ we obtain an asymptotic expansion of the form $\sum_{k=1}^n c_kx^{-k\alpha}+O(x^{-(n+1)\alpha})$, $x\to\infty$ ($n$ is determined by the smoothness of $h$). To establish the expansion we use a new approach which is based on the decomposition of the distribution into the sum of linear functionals.