Inequalities of correlation type for symmetric stable random vectors

Abstract

We prove that, for any jointly stable random variables $X_1, \dots, X_k$ with zero mean, any $m<k,$ and any even continuous positive definite functions $f$ and $g$ on $\Bbb R^m$ and $\Bbb R^{k-m},$ the random variables $f(X_1,\dots,X_m)$ and $g(X_{m+1}, \dots,X_k)$ are non-negatively correlated. We also show another result that is related to an old question of whether $$P(\max_{1\le i\le k} |X_i|<t) \ge P(\max_{1\le i\le m} |X_i|<t) \ P(\max_{m+1\le i\le k} |X_i|<t)$$ where $X_1,\dots,X_k$ are jointly Gaussian random variables with zero mean, and $m<k.$ We show that the quantity in the left-hand side has a local minimum at the point where the random variables $X_i$ and $X_j$ are independent for any choice of $1\le i\le m$ and $m+1\le j\le k.$