Bivariate Function Spaces and the Embedding of Their Marginal Spaces View Full Text


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Article Info

DATE

2006-08-09

AUTHORS

J. J. Grobler

ABSTRACT

.For a probability space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X\, \times \,Y,\,\Sigma \, \otimes \,\Lambda ,\,\user2{\mathbb{P}})$$\end{document} we denote the marginal measures of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\user2{\mathbb{P}}$$\end{document}, defined on Σ and Λ respectively, by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\user2{\mathbb{P}}_1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\user2{\mathbb{P}}_2$$\end{document}. If ρ is a function norm defined on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^0(X\, \times \,Y,\,\Sigma \, \otimes \,\Lambda ,\,\user2{\mathbb{P}})$$\end{document} marginal function norms ρ1 and ρ2 are defined on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^0 (X,\,\Sigma, \,\user2{\mathbb{P}}_1 )$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^0 (Y,\,\Lambda, \,\user2{\mathbb{P}}_2 )$$\end{document}. We find conditions which guarantee Lρ 1 + Lρ 2 to be embedded in Lρ as a closed subspace. The problem is encountered in Statistics when estimating a bivariate distribution with known marginals. We find a condition which, applied to the binormal distribution in L2, improves some known conditions. More... »

PAGES

83-99

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s00020-006-1447-z

DOI

http://dx.doi.org/10.1007/s00020-006-1447-z

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1011102729


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