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j \\le n.$\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 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correlation inequality for the expectations of norms of stable vectors","license":"","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Alexander Koldobsky","submitted_at":"1996-03-20T00:00:00Z","abstract_excerpt":"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 $1\\le i \\le k,\\ k+1\\le j \\le n.$\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 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