RANDOM DISCRIMINANTS

Let X1, X2,...,X(n) be a random sample from a continuous univariate distribution F, and let DELTA = PI 1 less-than-or-equal-to i < j less-than-or-equal-to n(X(i) - X(j))2 denote the t, or square of the Vandermonde determinant, constructed from the random sample. The statistic DELTA arises in the study of moment matrices and inference for mixture distributions, the spectral theory of random matrices, control theory and statistical physics. In this paper, we study the probability distribution of DELTA. When X1,...,X(n) is a random sample from a normal, gamma or beta population, we use Selberg's beta integral formula to obtain stochastic representations for the exact distribution of DELTA. Further, we obtain stochastic bound s for DELTA in the normal and gamma cases. Using the theory of U-statistics, we derive the asymptotic distribution of DELTA under certain conditions on F.