Sequential accumulation of independent cultural traits: A Gaussian limit for the popularity spectrum

Cultural traits, unlike genes, are acquired sequentially across and within individuals' lifetimes through repeated learning, innovation, and loss. While many tractable neutral models of cultural evolution assume one-shot acquisition, in which an individual's repertoire is fixed when it enters the population, we study a neutral model in which long-lived individuals in a fixed population repeatedly copy traits from one another and occasionally forget them, so that cultural turnover is driven by learning and forgetting within lifetimes rather than by births and deaths. The process admits a unique limiting distribution, with a stationary trait popularity spectrum in closed form. When the social learning efficiency is subcritical, the spectrum is monotone and total cultural diversity grows linearly with population size. Above the critical value the spectrum concentrates around an intermediate frequency and diversity grows exponentially with population size. Our main result concerns this supercritical regime: we prove that the spectrum is asymptotically Gaussian, with explicit mean, width, height and error bounds, and from this we derive the asymptotic behaviour of cultural diversity at the population and individual levels and of the time to stationarity. The same spectrum arises in the group-level model of Nakamura et al. (2020), so the shape of a spectrum does not by itself identify the mechanism behind it. Because the supercritical regime arises whenever individuals get more than one chance to acquire a trait, intermediate-peaked popularity distributions and abundant shared culture need not by themselves be read as evidence of biased or selective transmission.