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June, 1981 Operator-Stable Distribution on $R^2$ with Multiple Exponents
William N. Hudson, J. David Mason
Ann. Probab. 9(3): 482-489 (June, 1981). DOI: 10.1214/aop/1176994420


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 $R^2$.


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William N. Hudson. J. David Mason. "Operator-Stable Distribution on $R^2$ with Multiple Exponents." Ann. Probab. 9 (3) 482 - 489, June, 1981.


Published: June, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0465.60020
MathSciNet: MR614632
Digital Object Identifier: 10.1214/aop/1176994420

Primary: 60E05

Keywords: central limit theorem , multivariate stable laws , Operator-stable distributions

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 3 • June, 1981
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