The Annals of Probability

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

William N. Hudson and J. David Mason

Full-text: Open access

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


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