This is an announcement for the paper "Positive definite functions and multidimensional versions of random variables" by Alexander Koldobsky.

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.

Archive classification: math.PR math.FA

Mathematics Subject Classification: 60E05

The source file(s), posdef1.tex: 32112 bytes, is(are) stored in gzipped form as 0903.1433.gz with size 10kb. The corresponding postcript file has gzipped size 86kb.

Submitted from: koldobskiya@missouri.edu

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