2009/03/08 by Alexander Koldobsky, Koldobsky, Alexander · 1 citation
Mathematics · #Point processes and geometric inequalities #Stochastic processes and statistical mechanics #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.0903.1433
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 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.