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Positive Definite Distributions and Subspaces of L_p With Applications to Stable Processes

Published online by Cambridge University Press:  20 November 2018

Alexander Koldobsky
Alexander Koldobsky*
Affiliation:
Division of Mathematics and Statistics University of Texas at San Antonio San Antonio, Texas 78249 U.S.A., email: koldobsk@math.utsa.edu
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Abstract

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We define embedding of an $n$-dimensional normed space into ${{L}_{-p}},\,0\,<\,p\,<\,n$ by extending analytically with respect to $p$ the corresponding property of the classical ${{L}_{p}}$-spaces. The well-known connection between embeddings into ${{L}_{p}}$ and positive definite functions is extended to the case of negative $p$ by showing that a normed space embeds in ${{L}_{-p}}$ if and only if $\parallel x{{\parallel }^{-p}}$ is a positive definite distribution. We show that the technique of embedding in ${{L}_{-p}}$ can be applied to stable processes in some situations where standard methods do not work. As an example, we prove inequalities of correlation type for the expectations of norms of stable vectors. In particular, for every $P\in [n-3,n),\mathbb{E}({{\max }_{i=1,...,n}}{{\left| {{X}_{i}} \right|}^{-p}})\ge \mathbb{E}({{\max }_{i=1,...,n}}{{\left| {{Y}_{i}} \right|}^{-p}})$, where ${{X}_{1}},...,{{X}_{n}}\,\text{and}\,{{Y}_{1}},...,{{Y}_{n}}$ are jointly $q$-stable symmetric random variables, $0\,<\,q\,\le \,2$, so that, for some $k\,\in \,\mathbb{N},\,1\,\le \,k\,<\,n$, the vectors $\left( {{X}_{1}},\,.\,.\,.\,,\,{{X}_{k}} \right)$ and $\left( {{X}_{k+1}},\,.\,.\,.\,,{{X}_{n}} \right)$ have the same distributions as $({{Y}_{1}},...,{{Y}_{k}})\,\,\text{and}\,\,({{Y}_{k+1}},...,{{Y}_{n}})$, respectively, but ${{Y}_{i}}\,\text{and}\,{{Y}_{j}}$ are independent for every choice of $1\,\le \,i\,\le \,k,\,k\,+\,1\,\le \,j\,\le \,n$.

Keywords

42A8246B0446F1260E07
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 42 , Issue 3 , 01 September 1999 , pp. 344 - 353
Copyright
Copyright © Canadian Mathematical Society 1999

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