Elementary symmetric polynomials of increasing order

Abstract

Summary. The asymptotic behaviour of elementary symmetric polynomials S~k) of order k, based on n independent and identically distributed random variables X i. ... , Xn, is investigated for the case that both k and n get large.If k=Q(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-t), provided k= (O(log-1 nlog2 1 nn*) 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-+r::JJ at the rate nt 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--+ oo at a rate faster than n* then it is shown that for positive X/s there exists no linear norming which causes S~k> to converge weakly to a nondegenerate weak limit.

On a crossroad of resampling plans: bootstrapping elementary symmetric polynomials

“…This confirms Theorem I in VAN Es and HELMERS (1988). A detailed proof can be found in VAN Es et al (1997), an earlier version of this paper.…”

“…(13)) as Studentization by means of the delete-one-jackknife method, which is applied in HELMERS (1991) and MAESONO (1995). Combination of this fact with an argument like the one described in the appendix of VAN Es and HELMERS (1988) will then complete our proof. Similarly, one can also show that G~k)* (x) = <P(x) + ~n-112 <f>(x){(2x 2 + l)s~3m3 + 3(k -l)(x 2 + l)snk,;" 1 } + o(n~1 2 )…”

“…It can be proved by a slight adaptation of the proof given in MAESONO (1995) (cf. also HELMERS (1991) and VAN Es and HELMERS (1988)) that G~k)(x) = <P(x) + ~n-112 </J(x)…”

“…We adapt the proof of Theorem 1 in VAN Es and HELMERS (1988) to the bootstrap world. The proof is based on the Hoeffding decomposition of elementary symmetric polynomials, as given by KARLIN and RINOTI (1982).…”

“…For this case, the question how far the standard asymptotic normality, stated in (3), still holds if we allow k to increase with n, has been investigated in VAN Es and HELMERS (1988). It turns out that essentially we have to require k = o(n 1 1 2 ) for the polynomials to remain asymptotically normally distributed, with the standardization given by (2).…”