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,