Linkage Disequilibrium and the Infinitesimal Limit

Under the classical Fisher–Bulmer infinitesimal model of quantitative genetics, the within-family distribution for an additive trait with no environmental component is Gaussian with mean at the mid-parent value and a variance which is the same for all families. When an additive trait is determined by unlinked loci, the Fisher–Bulmer model can arise in the limit as the number of loci contributing to variation in the trait increases. However, a counterexample is presented where the Fisher–Bulmer model fails to arise in the infinite locus limit because there is too much linkage disequilibrium. An example is also presented where a degenerate form of the Fisher–Bulmer model arises. Under what conditions does the Fisher–Bulmer model arise in the infinite locus limit? It follows from the central limit theorem that the within-family distribution is Gaussian. But, under what conditions is the within-family distribution the same for almost all families in the population? An alternative population genetic derivation of the Fisher–Bulmer model is presented for a population at linkage equilibrium. This approach is then extended to allow many patterns of linkage disequilibrium. Diallelic models are used to illustrate the type of linkage disequilibrium allowed. The results on the limiting behaviour of population genetic models with many unlinked loci can be regarded as special cases of a more general limiting property of sequences of random variables. A possible application of this more general result to models of cultural inheritance is suggested.