Positive definite functions and stable random vectors

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 ΣaiXi and γ(a)Y are identically distributed, where γ: ℝn → [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 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.