On Convolutions and Linear Combinations of Pseudo-Isotropic Distributions

Roman Ger, Michael Keane and Jolanta K. Misiewicz
Additional contact information
Roman Ger: Uniwersytetu Śl¸askiego
Michael Keane: CWI
Jolanta K. Misiewicz: Politechniki Zielonogórskiej

Journal of Theoretical Probability, 2000, vol. 13, issue 4, 977-995

Abstract: Abstract Let E be a Banach space, and let E* be its dual. For understanding the main results of this paper it is enough to consider E= $$\mathbb{R}$$ n . A symmetric random vector X taking values in E is called pseudo-isotropic if all its one-dimensional projections have identical distributions up to a scale parameter, i.e., for every ξ∈E* there exists a positive constant c(ξ) such that (ξ, X) has the same distribution as c(ξ) X 0, where X 0 is a fixed nondegenerate symmetric random variable. The function c defines a quasi-norm on E*. Symmetric Gaussian random vectors and symmetric stable random vectors are the best known examples of pseudo-isotropic vectors. Another well known example is a family of elliptically contoured vectors which are defined as pseudo-isotropic with the quasi-norm c being a norm given by an inner product on E*. We show that if X and Y are independent, pseudo-isotropic and such that X+Y is also pseudo-isotropic, then either X and Y are both symmetric α-stable, for some α∈(0, 2], or they define the same quasi-norm c on E*. The result seems to be especially natural when restricted to elliptically contoured random vectors, namely: if X and Y are symmetric, elliptically contoured and such that X+Y is also elliptically contoured, then either X and Y are both symmetric Gaussian, or their densities have the same level curves. However, even in this simpler form, this theorem has not been proven earlier. Our proof is based upon investigation of the following functional equation: $$\begin{gathered} \exists m \in \left( {0,1} \right)\forall a,b >0,{a \mathord{\left/ {\vphantom {a {b \in \left[ {m,m^{ - 1} } \right]}}} \right. \kern \nulldelimiterspace} {b \in \left[ {m,m^{ - 1} } \right]}}\exists q\left( {a,b} \right) > 0\forall t > 0 \hfill \\\varphi \left( {at} \right)\psi \left( {bt} \right) = {\chi }\left( {q\left( {a,b} \right)t} \right) \hfill \\ \end{gathered}$$ which we solve in the class of real characteristic functions.

Keywords: convolutions; Banach space; Gaussian; pseudo-isotropic (search for similar items in EconPapers)
Date: 2000
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1023/A:1007809907072 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:13:y:2000:i:4:d:10.1023_a:1007809907072

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1023/A:1007809907072

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().