## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 3 (1981), 482-489.

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

William N. Hudson and J. David Mason

#### 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 $R^2$.

#### Article information

**Source**

Ann. Probab., Volume 9, Number 3 (1981), 482-489.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994420

**Digital Object Identifier**

doi:10.1214/aop/1176994420

**Mathematical Reviews number (MathSciNet)**

MR614632

**Zentralblatt MATH identifier**

0465.60020

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60E05: Distributions: general theory

**Keywords**

Operator-stable distributions multivariate stable laws central limit theorem

#### Citation

Hudson, William N.; Mason, J. David. Operator-Stable Distribution on $R^2$ with Multiple Exponents. Ann. Probab. 9 (1981), no. 3, 482--489. doi:10.1214/aop/1176994420. https://projecteuclid.org/euclid.aop/1176994420