Elementary Symmetric Polynomials of Increasing Order

摘要

The asymptotic behavior of elementary symmetric polynomials Sn of order k, based on n independent and identically distributed random variables X1, ..., Xn, is investigated for the case that both k and n get large. If k = 0micron (sq root n) the distribution function of a suitably normalized Sn of order k is shown to converge to a standard normal limit. The speed of this convergence to normality is of order 0 (k sq root n) if k = O (log (-1) n log2 (-1) n sq root n) and natural moment assumptions are imposed. This order bound is sharp, and cannot be inferred from one of the existing Berry-Esseen bounds for U-statistics. If k tends to infinity at the rate sq root n, then a non-normal weak limit appears, if the Xj's are positive and Sn(k) is standardized appropriately. If k tends to infinity at a rate faster than sq root n, it is shown that for positive Xj's there exists no linear norming which causes Sn(k) to converge weakly to a nondegenerate weak limit.