Operator-Stable Distribution on $R^2$ with Multiple Exponents

Abstract

Operator-stable distributions are the $n$-dimensional analogues of stable distributions when nonsingular matrices are used for scaling. Every full operator-stable distribution $\mu$ has an exponent, that is, a nonsingular linear transformation $A$ such that for every $t > 0 \mu^t = \mu t^{-A}\ast\delta(a(t))$ for some function $a: (0, \infty) \rightarrow R^n$. Full operator-stable distributions on $R^2$ have multiple exponents if and only if they are elliptically symmetric; in this case the characteristic functions are of the form $\exp\{iy'Vw - c|Vy|^\gamma\}$ where $V$ is positive-definite and self-adjoint, $0, < \gamma \leq 2, c > 0$, and $w$ is a point in $R^2$.