A correlation inequality for the expectations of norms of stable vectors

Abstract

For $0<q\le 2,\ 1\le k < n,$ let $X=(X_1,...,X_n)$ and $Y=(Y_1,...,Y_n)$ be symmetric $q$-stable random vectors so that the joint distributions of $X_1,...,X_k$ and $X_{k+1},...,X_n$ are equal to the joint distributions of $Y_1,...,Y_k$ and $Y_{k+1},...,Y_n,$ respectively, but $Y_i$ and $Y_j$ are independent for every $1\le i \le k,\ k+1\le j \le n.$ We prove that $\Bbb E (f(X)) \ge \Bbb E (f(Y))$ where $f$ is any continuous, positive, homogeneous of the order $p\in (-n,0)$ function on $\Bbb R^n\setminus \{0\}$ such that $f$ is a positive definite distribution in $\Bbb R^n,$ and $f(u,v)=f(u,-v)$ for every $u\in \Bbb R^k,\ v\in \Bbb R^{n-k}.$ As a particular case, we show that $$\Bbb E\ (\max_{i=1,...,n} |X_i|)^p \ge \Bbb E\ (\max_{i=1,...,n} |Y_i|)^p$$ for every $p\in (-n,-n+1).$ The latter inequality is related to Slepian&#39;s Lemma and to the Gaussian correlation problem.