Probability Laws with 1-Stable Marginals are 1-Stable

Abstract

We show that if $\mathbf{X} = (X_1,\ldots,X_d)$ is a vector in $\mathbb{R}^d$ and all linear combinations $\sum^d_{i=1}C_iX_i$ are 1-stable random variables, then $\mathbf{X}$ is itself 1-stable. More generally, a probability measure $\mu$ on a vector space whose univariate marginals are 1-stable is itself 1-stable. This settles an outstanding problem of Dudley and Kanter.