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On the Stability of One Characterization of Stable Distributions

On the Stability of One Characterization of Stable Distributions In 1923, G. Polya proved that if X 1 and X 2 are independent identically distributed random variables (i.i.d.r.v.) with finite variance, then the distributions of X 1 and (X 1+X 2)/ $$\sqrt 2$$ are coincidental iff X 1 has the normal distribution with zero mean. Is an analogous theorem possible for an couple of statistics X 1 and (X 1+X 2)/21/α if α<2? P. Lévy constructed an example that denies that hypothesis. However, having supplemented the condition of coincidence of the distributions of X 1 and (X 1+X 2)/21/α with a similar condition, namely, requiring, in addition, for the distributions of X 1 and (X 1+X 2+X 3)/31/α to be coincident (here X 1,X 2 and X 3 are i.i.d.r.v.), P. Lévy has proved that X 1 and X 2 have a strictly stable distribution. The stability of this characterization in a metric λ0 (that is defined in the class of characteristic functions by analogy with a uniform metric defined in the class of distributions) without an additional symmetry assumption as well as the stability in a Lévy metric L are analizied in this paper. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

On the Stability of One Characterization of Stable Distributions

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References (6)

Publisher
Springer Journals
Copyright
Copyright © 2003 by Kluwer Academic Publishers
Subject
Mathematics; Mathematics, general; Computer Science, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Classical Mechanics
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1023/A:1025899831257
Publisher site
See Article on Publisher Site

Abstract

In 1923, G. Polya proved that if X 1 and X 2 are independent identically distributed random variables (i.i.d.r.v.) with finite variance, then the distributions of X 1 and (X 1+X 2)/ $$\sqrt 2$$ are coincidental iff X 1 has the normal distribution with zero mean. Is an analogous theorem possible for an couple of statistics X 1 and (X 1+X 2)/21/α if α<2? P. Lévy constructed an example that denies that hypothesis. However, having supplemented the condition of coincidence of the distributions of X 1 and (X 1+X 2)/21/α with a similar condition, namely, requiring, in addition, for the distributions of X 1 and (X 1+X 2+X 3)/31/α to be coincident (here X 1,X 2 and X 3 are i.i.d.r.v.), P. Lévy has proved that X 1 and X 2 have a strictly stable distribution. The stability of this characterization in a metric λ0 (that is defined in the class of characteristic functions by analogy with a uniform metric defined in the class of distributions) without an additional symmetry assumption as well as the stability in a Lévy metric L are analizied in this paper.

Journal

Acta Applicandae MathematicaeSpringer Journals

Published: Oct 18, 2004

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