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**1 - 1**of**1**### POSITIVE DEFINITE FUNCTIONS AND MULTIDIMENSIONAL VERSIONS OF RANDOM VARIABLES

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"... Abstract. We say that a random vector X = (X1,..., Xn) in R n is an n-dimensional version of a random variable Y if for any a ∈ R n the random variables ∑ aiXi and γ(a)Y are identically distributed, where γ: R n → [0, ∞) is called the standard of X. An old problem is to characterize those functions ..."

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Abstract. We say that a random vector X = (X1,..., Xn) in R n is an n-dimensional version of a random variable Y if for any a ∈ R n the random variables ∑ aiXi and γ(a)Y are identically distributed, where γ: R n → [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 n, where K is an origin symmetric star body in R n and f: R → R is an even continuous function, then either the space (R n, ‖ · ‖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. 1.