Elementary symmetric polynomials of increasing order

✍ Scribed by A. J. Es; R. Helmers

• Publisher: Springer
• Year: 1988
• Tongue: English
• Weight: 563 KB
• Volume: 80
• Category: Article
• ISSN: 1432-2064
• DOI: 10.1007/bf00348750

✦ Synopsis

The asymptotic behaviour of elementary symmetric polynomials S~, k) of order k, based on n independent and identically distributed random variables X1, ..., X,, is investigated for the case that both k and n get large.

If k=v(n~), then the distribution function of a suitably normalised S~, k) is shown to converge to a standard normal limit. The speed of this convergence to normality is of order (9(kn-~), provided k = (9(log-~ n log~ in n ~) and certain 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 ~ oo at the rate n ~ then a non-normal weak limit appears, provided the X{s are positive and S~ k) is standardised appropriately. On the other hand, if k---, oe at a rate faster than n ~-then it is shown that for positive Xjs there exists no linear norming which causes S~ ) to converge weakly to a nondegenerate weak limit.