Random Discriminants

Abstract

Let $X_1, X_2, \cdots, X_n$ be a random sample from a continuous univariate distribution $F$, and let $\Delta = \prod_{1 \leq i < j \leq n}(X_i - X_j)^2$ denote the discriminant, 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 $X_1, \cdots, X_n$ 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 bounds 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 $F$.