A Correlation Inequality for the Expectations of Norms of Stable Vectors

For 0 < q ≤ 2, 1 ≤ k < n, let X = (X1, ...,Xn) and Y = (Y1, ..., Yn) be symmetric q-stable random vectors so that the joint distributions of X1, ...,Xk and Xk+1, ...,Xn are equal to the joint distributions of Y1, ..., Yk and Yk+1, ..., Yn, respectively, but Yi and Yj are independent for every 1 ≤ i ≤ k, k + 1 ≤ j ≤ n. We prove that E(f(X)) ≥ E(f(Y )) where f is any continuous, positive, homogeneous of the order p ∈ (−n, 0) function on R\{0} such that f is a positive definite distribution in R, and f(u, v) = f(u,−v) for every u ∈ R, v ∈ R. As a particular case, we show that E ( max i=1,...,n |Xi|) p ≥ E ( max i=1,...,n |Yi|) p for every p ∈ (−n,−n+ 1). The latter inequality is related to Slepian’s Lemma and to the Gaussian correlation problem.