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In this paper we explore the theoretical limits of the inference of cultural transmission modes based on sparse population-level data. We approach this problem by investigating whether different transmission modes produce different temporal dynamics of cultural change. In particular we explore whether the distributions of the average time a variant stays the most common variant in the population, denoted by tmax, conditioned on the considered transmission modes are sufficiently different to allow for inference of underlying transmission modes. We assume time series data detailing the frequencies of different variants of a cultural trait in a population at different points in time and investigate the temporal resolution (i.e. the length of the time series and the distance between consecutive time points) that is needed to ensure distinguishability between transmission modes. We find that under complete information most transmission modes can be distinguished on the base of the statistic tmax, however we should not expect the same results if only infrequent information about the most common cultural variant in the population are available.