A representation theorem applied to weakly exchangeable non-negative definite arrays
Let \(X=(X_{ij})_{1\leq i,j<\infty}\) be a weakly exchangeable symmetric non-negative definite (WESND) array, i.e. a random real valued array such that: (i) \((X_{ij})_{1\leq i,j<\infty}=^{d}(X_{\pi i,\pi j})_{1\leq i,j<\infty}\) for any permutation \(\pi\) that leaves all but a finite number of integers fixed; (ii) \(X_{ij}=X_{ji}\) for all i and j; and (iii) the matrix \((X_{ij})_{1\leq i,j\leq m}\) is non-negative definite for each \(m\geq 1.\) Assuming that E \(X^ 2_{ii}<\infty\) for all i, the author shows that the distribution of X is extreme if and only if \(X=^{d}Y+Z\), where Y is a WESND diagonal array with i.i.d. non-negative diagonal entries, and Z is a WESND array whose entries are inner products of i.i.d. random elements from a Hilbert space.
scientific article; zbMATH DE number 4138867