Section 2.1.4.5. Non-independent variables

The central-limit theorem, under certain conditions, remains valid even when the variables summed in equation (2.1.4.19) are not independent. The conditions have been investigated by Bernstein (1922, 1927); roughly they amount to requiring that the variables should not be too closely correlated. The theorem applies, in particular, when each is related to a finite number, , of its neighbours, when the x's are said to be dependent. The dependence seems plausible for crystallographic applications, since the positions of atoms close together in a structure are closely correlated by interatomic forces, whereas those far apart will show little correlation if there is any flexibility in the asymmetric unit when unconstrained. Harker's (1953) idea of `globs' seems equivalent to dependence. Long-range stereochemical effects, as in pseudo-graphitic aromatic hydrocarbons, would presumably produce long-range correlations and make dependence less plausible. If Bernstein's conditions are satisfied, the central-limit theorem would apply, but the actual value of would have to be used for the variance, instead of the sum of the variances of the random variables in (2.1.4.19). Because of the correlations the two values are no longer equal.

French & Wilson (1978) seem to have been the first to appeal explicitly to the central-limit theorem extended to non-independent variables, but many previous workers [for typical references, see Wilson (1981)] tacitly made the replacement – in the X-ray case substituting the local mean intensity for the sum of the squares of the atomic scattering factors.