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. In addition, we study the intergenerational transmission of the sex ratio in models of threshold (e. g. , polygenic or temperature-dependent) sex determination with long-tailed sex-determining traits. Among other results, we show that if the distribution of the sex determining trait is not very thick-tailed, then several properties of these models are the same as in the case of log-concave densities analyzed by Karlin (1984, 1992). In particular, the excess of males (females) among parents leads to the same pattern for the population of the offspring. Thus, the excess of one sex over the other one accumulates with time and the sex ratio in the total alive population cannot stabilize at the balanced sex ratio value of 1/2. We further show that the above properties are reversed for sufficiently heavy-tailed distributions of sex determining traits. In such settings, the sex ratio of the offspring oscillates around the balanced sex ratio value and an excess of males (females) in the initial period leads to an excess of females (males) offspring next period. Therefore, the sex ratio in the total living population can, in fact, stabilize at 1/2. Interestingly, these results are related, in particular, to the analysis of correlation between human sex ratios and socioeconomic status of parents as well as to the study of the variation of the sex ratio due to parental hormonal levels. 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 (2004) and several their extensions derived in this work.