Some highlights from the theory of multivariate symmetries

We explain how invariance in distribution under separate or joint contractions, permutations, or rotations can be defined in a natural way for d-dimensional arrays of random variables. In each case, the distribution is characterized by a general representation formula, often easy to state but surpr...

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Autor principal:	Olav Kallenberg
Formato:	article
Lenguaje:	EN FR IT
Publicado:	Sapienza Università Editrice 2008
Materias:	multiple wiener-itô integrals contractable exchangeable and rotatable random arrays functionals and sheets Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/d643af75d30b4823837b1415f06810ab
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id	oai:doaj.org-article:d643af75d30b4823837b1415f06810ab
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spelling	oai:doaj.org-article:d643af75d30b4823837b1415f06810ab2021-11-29T14:14:26ZSome highlights from the theory of multivariate symmetries1120-71832532-3350https://doaj.org/article/d643af75d30b4823837b1415f06810ab2008-01-01T00:00:00Zhttps://www1.mat.uniroma1.it/ricerca/rendiconti/ARCHIVIO/2008(1)/19-32.pdfhttps://doaj.org/toc/1120-7183https://doaj.org/toc/2532-3350We explain how invariance in distribution under separate or joint contractions, permutations, or rotations can be defined in a natural way for d-dimensional arrays of random variables. In each case, the distribution is characterized by a general representation formula, often easy to state but surprisingly complicated to prove. Comparing the representations in the first two cases, one sees that an array on a tetrahedral index set is contractable iff it admits an extension to a jointly exchangeable array on the full rectangular index set. Multivariate rotatability is defined most naturally for continuous linear random functionals on tensor products of Hilbert spaces. Here the simplest examples are the multiple Wiener–Itô integrals, which also form the basic building blocks of the general representations. The rotatable theory can be used to derive similar representations for separately or jointly exchangeable or contractable random sheets. The present paper provides a non-technical survey of the mentioned results, the complete proofs being available elsewhere. We conclude with a list of open problems.Olav KallenbergSapienza Università Editricearticlemultiple wiener-itô integralscontractableexchangeable and rotatable random arraysfunctionals and sheetsMathematicsQA1-939ENFRITRendiconti di Matematica e delle Sue Applicazioni, Vol 28, Iss 1, Pp 19-32 (2008)
institution	DOAJ
collection	DOAJ
language	EN FR IT
topic	multiple wiener-itô integrals contractable exchangeable and rotatable random arrays functionals and sheets Mathematics QA1-939
spellingShingle	multiple wiener-itô integrals contractable exchangeable and rotatable random arrays functionals and sheets Mathematics QA1-939 Olav Kallenberg Some highlights from the theory of multivariate symmetries
description	We explain how invariance in distribution under separate or joint contractions, permutations, or rotations can be defined in a natural way for d-dimensional arrays of random variables. In each case, the distribution is characterized by a general representation formula, often easy to state but surprisingly complicated to prove. Comparing the representations in the first two cases, one sees that an array on a tetrahedral index set is contractable iff it admits an extension to a jointly exchangeable array on the full rectangular index set. Multivariate rotatability is defined most naturally for continuous linear random functionals on tensor products of Hilbert spaces. Here the simplest examples are the multiple Wiener–Itô integrals, which also form the basic building blocks of the general representations. The rotatable theory can be used to derive similar representations for separately or jointly exchangeable or contractable random sheets. The present paper provides a non-technical survey of the mentioned results, the complete proofs being available elsewhere. We conclude with a list of open problems.
format	article
author	Olav Kallenberg
author_facet	Olav Kallenberg
author_sort	Olav Kallenberg
title	Some highlights from the theory of multivariate symmetries
title_short	Some highlights from the theory of multivariate symmetries
title_full	Some highlights from the theory of multivariate symmetries
title_fullStr	Some highlights from the theory of multivariate symmetries
title_full_unstemmed	Some highlights from the theory of multivariate symmetries
title_sort	some highlights from the theory of multivariate symmetries
publisher	Sapienza Università Editrice
publishDate	2008
url	https://doaj.org/article/d643af75d30b4823837b1415f06810ab
work_keys_str_mv	AT olavkallenberg somehighlightsfromthetheoryofmultivariatesymmetries
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Some highlights from the theory of multivariate symmetries

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