Analyzing the rate at which languages change can clarify whether similarities across languages are solely the result of cognitive biases or might be partially due to descent from a common ancestor. To demonstrate this approach, we use a simple model of language evolution to mathematically determine how long it should take for the distribution over languages to lose the influence of a common ancestor and converge to a form that is determined by constraints on language learning. We show that modeling language learning as Bayesian inference of n binary parameters or the ordering of n constraints results in convergence in a number of generations that is on the order of n log n. We relax some of the simplifying assumptions of this model to explore how different assumptions about language evolution affect predictions about the time to convergence; in general, convergence time increases as the model becomes more realistic. This allows us to characterize the assumptions about language learning (given the models that we consider) that are sufficient for convergence to have taken place on a timescale that is consistent with the origin of human languages. These results clearly identify the consequences of a set of simple models of language evolution and show how analysis of convergence rates provides a tool that can be used to explore questions about the relationship between accounts of language learning and the origins of similarities across languages.