Consistency of the maximum likelihood estimator of population tree in a coalescent framework

We present a proof of consistency of the maximum likelihood estimator (MLE) of population tree in a previously proposed coalescent model. As the model involves tree-topology as a parameter, the standard proof of consistency for continuous parameters does not directly apply. In addition to proving that a consistent sequence of MLE exists, we also prove that the overall MLE, computed by maximizing the likelihood over all tree-topologies, is also consistent. Thus, the MLE of tree-topology is consistent as well. The last result is important because local maxima occur in the likelihood of population trees, especially while maximizing the likelihood separately for each tree-topology. Even though MLE is known to be a dependable estimator under this model, our work proves its effectiveness with mathematical certainty.