Asymptotic estimates for densities of multi-dimensional stable distributions
Seiji Hiraba · Tsukuba Journal of Mathematics · 2003
Introduction and ResultsLet $\mu(dx)$ be a stable distribution on $R^{d}$ with exponent $00),$ $=0(x=0),$ $=-1(x<0),$ $\lambda(d\theta)$ is a finite measure on $S^{d-1}$ and $b\in R^{d}$ .Moreover if $\mu$ is non-degenerate, then $\mu$ has a $C^{\infty_{-}}$ density function $p(x)$ with respect to the Lebesgue measure $dx$ , i.e., $\mu(dx)=$ $p(x)dx$ and(1.1) $p(x)=\frac{1}{(2\pi)^{d}}\int_{R^{d}}\exp[-i\langle x, z\rangle+\Psi(z)]dz$ .The non-degeneracy of $\mu$ means $SpanSpt\mu=R^{d}$ and it is equivalent to Span Spt $\lambda=R^{d}$ , where Spt $\mu$ (resp.Spt $\lambda$ ) is a support of $\mu$ (resp.$\lambda$ ) and for a set $S\subset R^{d}$ , Span $S$ is a linear subspace of $R^{d}$ spanned by $S$ (cf. [3]).In the present paper we would like to investigate the asymptotic behavior of $p(r\sigma)$ as $ r\rightarrow\infty$ for each direction $\sigma\in S^{d-1}$ under the following assumption.ASSUMPTION 1.Let $b=0$ .For some number $m\geq 0$ , $Spt\lambda=\{\sigma^{(1)}, \sigma^{(2)}, \ldots, \sigma^{(d+m)}\}\subset S^{d-1}$ and $SpanSpt\lambda=R^{d}$ , that is, the support of $\lambda$ is only finitely many points which linearly spans $R^{d}$ .