The reconstruction of phylogenies from DNA or protein sequences is a major task of computational evolutionary biology. Common phenomena, notably variations in mutation rates across genomes and incongruences between gene lineage histories, often make it necessary to model molecular data as originating from a mixture of phylogenies. Such mixed models play an increasingly important role in practic...

1.1 Concentration of measure: examples We start with some examples of “concentration of measure”. This phenomenon is very useful in analyzing machine learning algorithms and can be used to bound things like error probabilities. The first and the most standard of concentration results is for averages. It states that “the average of bounded independent random variables is tightly concentrated aro...

Over the past decade there has been a flurry of new concentration of measure inequalities; we refer the reader to [4] for an in-depth survey, or [2, 3, 5] for some more recent advances. In [2] the martingale difference method was employed in a novel way to obtain a general concentration inequality for dependent random variables, with respect to the (unweighted) Hamming metric. At the core of th...

We derive sufficient conditions for a family (Sn, ρn,Pn) of metric probability spaces to have the measure concentration property. Specifically, if the sequence {Pn} of probability measures satisfies a strong mixing condition (which we call η-mixing) and the sequence of metrics {ρn} is what we call Ψ-dominated, we show that (Sn, ρn,Pn) is a normal Lévy family. We establish these properties for s...