We say that a random vector X=(X1, ..., Xn) in Rn is an n-dimensional version of a random variable Y if for any a ? Rn the random variables ?aiXi and g(a) Y are identically distributed, where g:Rn? [0, 8) is called the standard of X. An old problem is to characterize those functions g 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 L0. This result is almost optimal, as the norm of any finite dimensional subspace of Lp with p ? (0, 2] is the standard of an n-dimensional version (p-stable random vector) by the classical result of P.Lèvy. An equivalent formulation is that if a function of the form f(?·?K) is positive definite on Rn, where K is an origin symmetric star body in Rn and f:R? R is an even continuous function, then either the space (Rn, ?·?K) embeds in L0 or f is a constant function. Combined with known facts about embedding in L0, this result leads to several generalizations of the solution of Schoenberg's problem on positive definite functions.