A tale of two tails: peakedness properties in inheritance models of evolutionary theory

In this paper, we study transmission of traits through generations in multifactorial inheritance models with sex- and time-dependent heritability. We further analyze the implications of these models under heavy-tailedness of traits' distributions. Among other results, we show that in the case of a trait (for instance, a medical or behavioral disorder or a phenotype with significant heritability affecting human capital in an economy) with not very thick-tailed initial density, the trait distribution becomes increasingly more peaked, that is, increasingly more concentrated and unequally spread, with time. But these patterns are reversed for traits with sufficiently heavy-tailed initial distributions (e.g., a medical or behavioral disorder for which there is no strongly expressed risk group or a relatively equally distributed ability with significant genetic influence). Such traits' distributions become less peaked over time and increasingly more spread in the population. The proof of the results in the paper is based on the general results on majorization properties of heavy-tailed distributions obtained recently in Ibragimov (Econom Theory 23: 501-517, 2007) and also presented in the author's Ph.D. dissertation (Ibragimov, New majorization theory in economics and martingale convergence results in econometrics. Yale University, 2005) and several their extensions derived in this work.