Positive definite functions and multidimensional versions of random variables

TL;DR

This paper characterizes functions that serve as standards for multidimensional random variables, linking them to norms of spaces embedding in L0, and extends classical results on positive definite functions.

Contribution

It proves Lisitsky's conjecture that standards must be norms of spaces embedding in L0, and relates positive definite functions to embeddings in L0, generalizing Schoenberg's problem.

Findings

Every standard must be a norm of a space embedding in L0.

Any positive definite function of the form f(||·||_K) implies the space embeds in L0 or f is constant.

Classical p-stable vectors are examples of n-dimensional versions for p in (0,2].

Abstract

We say that a random vector $X=(X_1,...,X_n)$ in $R^n$ is an $n$-dimensional version of a random variable $Y$ if for any $a\in R^n$ the random variables $\sum a_iX_i$ and $\gamma(a) Y$ are identically distributed, where $\gamma:R^n\to [0,\infty)$ is called the standard of $X.$ An old problem is to characterize those functions $\gamma$ that can appear as the standard of an $n$-dimensional version. In this paper, we prove the conjecture of Lisitsky that every standard must be the norm of a space that embeds in $L_0.$ This result is almost optimal, as the norm of any finite dimensional subspace of $L_p$ with $p\in (0,2]$ is the standard of an $n$-dimensional version ($p$-stable random vector) by the classical result of P.L\`evy. An equivalent formulation is that if a function of the form $f(\|\cdot\|_K)$ is positive definite on $R^n,$ where $K$ is an origin symmetric star body in $R^n$ and $f:R\to R$ is an even continuous function, then either the space $(R^n,\|\cdot\|_K)$ embeds in $L_0$ or $f$ is a constant function. Combined with known facts about embedding in $L_0,$ this result leads to several generalizations of the solution of Schoenberg's problem on positive definite functions.