POSITIVE DEFINITE DISTRIBUTIONS AND SUBSPACES OF L−p WITH APPLICATIONS TO STABLE PROCESSES

We define embedding of an n-dimensional normed space into L−p, 0 < p < n by extending analytically with respect to p the corresponding property of the classical Lp-spaces. The well-known connection between embeddings into Lp and positive definite functions is extended to the case of negative p by showing that a normed space embeds in L−p if and only if ‖x‖ is a positive definite distribution. Using this criterion, we generalize the recent solutions to the 1938 Schoenberg’s problems by proving that the spaces lq , 2 < q ≤ ∞ embed in L−p if and only if p ∈ [n− 3, n). We show that the technique of embedding in L−p can be applied to stable processes in some situations where standard methods do not work. As an example, we prove inequalities of correlation type for the expectations of norms of stable vectors. In particular, for every p ∈ [n−3, n), E(maxi=1,...,n |Xi| ) ≥ E(maxi=1,...,n |Yi| ), where X1, ...,Xn and Y1, ..., Yn are jointly q-stable symmetric random variables, 0 < q ≤ 2, so that, for some k ∈ N, 1 ≤ k < n, the vectors (X1, ...,Xk) and (Xk+1, ...,Xn) have the same distributions as (Y1, ..., Yk) and (Yk+1, ..., Yn), respectively, but Yi and Yj are independent for every choice of 1 ≤ i ≤ k, k + 1 ≤ j ≤ n.