The Annals of Statistics

Random Discriminants

I-Li Lu and Donald Richards

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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 $F$.

Article information

Ann. Statist., Volume 21, Number 4 (1993), 1982-2000.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62E15: Exact distribution theory
Secondary: 62H10: Distribution of statistics 60E15: Inequalities; stochastic orderings 62H05: Characterization and structure theory 62G30: Order statistics; empirical distribution functions

Moment matrices mixture distributions random discriminants Selberg's beta integral stochastic bounds $U$-statistics Vandermonde determinants


Lu, I-Li; Richards, Donald. Random Discriminants. Ann. Statist. 21 (1993), no. 4, 1982--2000. doi:10.1214/aos/1176349406.

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