  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.

Probability;Functional Analysis