Positive definite distributions and subspaces of $L_{-p}$ with applications to stable processes

Abstract

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 $\Vert x{\Vert }^{-p}$ is a positive definite distribution. Using this criterion, we generalize the recent solutions to the 1938 Schoenberg's problems by proving that the spaces ${\ell }_q^n,2<q\le \infty$ embed in $L_{-p}$ if and only if $p\in [n-3,n).$ 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 $p\in [n-3,n),$ $\mathbb{E}({\max }_{i=1,...,n}|X_i{|}^{-p})\ge \mathbb{E}({\max }_{i=1,...,n}|Y_i{|}^{-p}),$ where $X_1,...,X_n$ 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 $(X_1,...,X_k)$ and $(X_{k+1},...,X_n)$ have the same distributions as $(Y_1,...,Y_k)$ and $(Y_{k+1},...,Y_n),$ respectively, but $Y_i$ and $Y_j$ are independent for every choice of $1\le i\le k,k+1\le j\le n.$

References in corpus (1)

• On the Gaussian measure of the intersection of symmetric, convex sets

Cited by in corpus (10)

• Intersection Bodies with Certain Symmetries
• A note on the Busemann-Petty problem for bodies of certain invariance
• The complex Busemann-Petty problem on sections of convex bodies
• Centroids and comparison of volumes
• Inequalities for the derivatives of the Radon transform on convex bodies
• Non-intersection bodies all of whose central sections are intersection bodies
• Second derivative test for isometric embeddings in $L_p$
• Schoenberg's Theorem via the law of large numbers
• On strict inclusions in hierarchies of convex bodies
• Isomorphic properties of Intersection bodies