On the computation of rational approximations to continuous functions
William Fraser, John Hart · Communications of the ACM · 1962
Asymptotic NormalityFor each t, x(t) in Eq. ( 6) is composed of a linear combination of identically and independently distributed random variables.It might be suspected that as N ~ the distribution of x(t) approaches a normal distribution with mean-zero and variance equal to R(0).Care must be exercised, however, since the A,~ coefficients are functions of N, as evidence by Eqs.(5).In other words, each case must be analyzed individually, as R(r) = cos r provides an example of a case in which the central limit theorem does not hold.The authors have tested a few cases, using a chi-squared goodness-of-fit test, and obtained results which indicate that when the central limit theorem applies, convergence is quite rapid.For example, let and let R(T) = 1 --i r l I r l =< 1 Rk = 0.02/~ 0 =< /c =< 50.A very small, chi-squared value with 9 degrees of freedom indicated normality at the 99 percent level of confidence.