Two multivariate central limit theorems are proved for polygenic trait values over a pedigree or collection of pedigrees. These theorems presuppose Hardy-Weinberg and linkage equilibrium for all loci, absence of assortative mating and epistasis, and a small variance for each locus compared to the total variance over many loci. To prevent clustering of loci, an upper bound is also imposed on the number of loci per chromosome. In the case of zero dominance variance for each locus, arbitrary inbreeding is allowed. Otherwise inbreeding is not permitted. Further technical conditions are spelled out in the text.