How long a newly emerging trait will stay in a population is a fundamental but rarely asked question in cultural evolution. To tackle this question, the distribution and mean of the time to extinction of a discrete cultural trait are derived for models with overlapping generations, in which trait transmission occurs from multiple role models to a single newborn and may fail with a certain probability. We explore two models. The first is a Moran-type model, which allows us to derive the exact analytical formula for the mean time to extinction of a trait in a finite population. The second is a branching process, which assumes an infinitely large population and allows us to derive approximate analytical formulae for the distribution and mean of the time to extinction in the first model under a large population size. We show that in the first model, the mean time to extinction apparently diverges (becomes so large that even numerical computation is impractical) under a certain parameter condition as the population size tends to infinity. Using the second model, we explain the underlying mechanism of the apparent divergence found in the first model and derive the mathematical condition for this divergence in terms of transmission efficiency and the number of role models per newborn. When this mathematical condition is satisfied in the second model, the probability of extinction is less than 1, and the mean extinction time does not exist. In addition, we find that in both models, the time to extinction of the trait becomes longer as the number of role models per individual increases and as cultural transmission becomes more efficient.

一个新出现的特征将在人口中保留多长时间是文化进化中的一个基本但很少问到的问题。为了解决这个问题，对于具有重叠世代的模型，推导出了离散文化特征消失的时间的分布和平均值，其中特征传播是从多个角色模型传播到单个新生儿，并且可能以一定的概率失败。我们探索两种模型。第一个是Moran型模型，它使我们能够得出有限种群中某​​性状消失的平均时间的精确解析公式。第二个过程是分支过程，它假设种群数量无限大，并允许我们导出在种群较大的情况下第一个模型中灭绝时间的分布和均值的近似分析公式。我们证明在第一个模型中 在一定的参数条件下，随着种群数量趋于无穷大，灭绝的平均时间明显发散（变得如此之大，以至于无法进行数值计算）。使用第二个模型，我们解释了第一个模型中发现的表观差异的潜在机制，并根据传播效率和每个新生儿的榜样数量得出了这种差异的数学条件。在第二个模型中满足此数学条件时，灭绝的概率小于1，并且平均灭绝时间不存在。此外，我们发现，在两种模式中，随着每个人的榜样数量增加以及文化传播变得更加有效，特征消失的时间会更长。