Positive Definite Functions and Multidimensional Versions of Random Variables

aiXi and γ(a)Y are identically distributed, where γ : R → [0,∞) is called the standard of X. An old problem is to characterize those functions γ 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 R , where K is an origin symmetric star body in R and f : R → R is an even continuous function, then either the space (R, ‖ · ‖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.