Asymptotic Behaviour of Symmetric Polynomial Statistics

Abstract

A general theorem of Hoeffding (Hoeffding (1948)) implies that the limit distribution of symmetric statistics under natural conditions is normal, but rather frequently the variance of this normal distribution vanishes. Even in the very simple case when the statistics are elementary symmetric polynomials of independent random variables $\{X_i: P(X_i = \pm 1) = \frac{1}{2}, i = 1,2, \cdots, n\}$, the limit distributions (in general) are not normal. These distributions will be determined below.