A CONSISTENCY LEMMA IN STATISTICAL PHYLOGENETICS
= T). Let θbe a vector of continuous parameters – including the branch lengthsof T, along with possibly other continuous parameters required to specify a model of characterevolution on T. Let Θ denote the set of values θ may take. Branch lengths, in particular,are assumed to be strictly positive and ﬁnite; and in general Θ will be some open subset ofEuclidean space. Consider any stochastic process (e.g. Markov process, or mixture of Markovprocesses) which assigns to each pair (T,θ) a probability distribution s = s(T,θ) on discrete,ﬁnite-state characters at the tips of the tree. We assume throughout that the map θ→ s(T,θ) iscontinuous. Such models are central to statistical phylogenetics and methods for reconstructingphylogenetic trees from aligned genetic (e.g. DNA) sequences. A tree reconstruction method ψisany method that reconstructs a set of one or more unrooted phylogenetic trees from any givendistribution ˆf of site pattern frequencies. Suppose we generate ksites i.i.d. from (T,θ), and let ˆsbe the random variable equal to the resulting proportion of site patterns (character types). Themethod ψis a statistically consistent estimator of the unrooted topology of Tif the probabilitythat ψ(ˆs) = {T