On the joint asymptotic distribution of additive genotype for polygenic characters (Q1076638)

In the Appendix of his important paper ''Selection and genetic variability'', The American Naturalist 105, 201-211 (1971) and again in chapter 8 of his book ''The mathematical theory of quantitative genetics'', Oxford (1980; Zbl 0441.92007), \textit{M. G. Bulmer} discusses the asymptotic distribution of a single quantitative character for two related individuals chosen at random from the population. The character is assumed to be under the control of many loci and the basic conclusion is that the bivariate central limit theorem applies at equilibrium when, effectively, loci behave independently. However, when linkage effects destroy this independence relationship between loci, a major condition of the theorem is violated and the resulting joint distribution is not normal. This is a very disturbing assertion since the normal distribution, in all its forms, is fundamental to most work in quantitative genetics. Further, it is counter-intuitive. The difficulty seems to be that the assumption of independence between bivariate pairs, which is a sufficient condition of the central limit theorem, is used as a necessary condition. In the following sections I will use results on m-dependence to show that, for a polygenic character under panmixia, the joint distribution of this character among related individuals can be anticipated to be normal under very general conditions.