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.
We investigate the validity of the bootstrap method for the elementaryrandom variables X 1 , ... ,Xn. For both fix~d and'incre~sing brder k, as n __, oo the cases where µ = EX 1 f= 0, the nondegenerate case, and where µ = EX 1 = 0, the degenerate case, are considered.
Some theory of linear congruential pseudo random number generators x , +~ = (ux, + c) mod m is summarized for the case in which the modulus m is a power of 10. These generators are especially suitable for implementation on both programmable and non-programmable pocket calculators. Results are presented of extensive statistical testing of two specifc generators,
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.