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

TL;DR

This paper extends the theory of embeddings of normed spaces into negative p-Lp spaces, linking positive definite distributions to stable processes, and applies these insights to derive inequalities for stable vectors.

Contribution

It introduces a novel criterion for embedding normed spaces into L_{-p} using positive definite distributions and applies this to solve classical problems and analyze stable processes.

Findings

Spaces ll_q^n embed in L_{-p} iff p  in [n-3,n)

Embedding techniques apply to derive inequalities for stable vectors

Established a connection between embeddings and positive definite distributions

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 $\|x\|^{-p}$ 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 $p\in [n-3,n),$ $\Bbb E(\max_{i=1,...,n} |X_i|^{-p}) \ge \Bbb E(\max_{i=1,...,n} |Y_i|^{-p}),$ 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 $k\in \Bbb N,\ 1\le k <n,$ 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 $1\le i\le k,\ k+1\le j\le n.$