A characterization of multivariate \(l_ 1\)-norm symmetric distributions (Q1117637)

Let z be an \(n\times 1\) interchangeable random vector and \(z_{(1)}\leq...\leq z_{(n)}\) be its order statistics. Let \[ u_ i=(n- i+1)(z_{(i)}-z_{(i-1)}),\quad i=1,...,n,\quad with\quad z_{(0)}=0 \] and \(u=(u_ 1,...,u_ n)'\). The main result is that \(z=^{d}u\) iff z is a multivariate \(\ell_ 1\)-norm symmetric distribution.