Phylogenetic dynamics of MRCA ages and empirical moments of a Brownian trait
We study the temporal dynamics of the first two empirical moments of Brownian traits on phylogenetic trees. For a fixed tree, we characterize the distributions of their empirical mean and empirical variance across all lineages extant at any given time. In particular, we show that the variance of the empirical mean and the expected empirical variance are piecewise linear between diversification events. For lineage-homogeneous random trees, both the variance of the empirical mean and the expected empirical variance can be expressed in terms of the expected age of the most recent common ancestor (MRCA) of a uniformly sampled pair of extant lineages. In this representation, the expected MRCA age enters the two quantities with opposite signs, pointing to a structural opposition between the variance of the empirical mean and the expected empirical variance. For generalized birth-death processes with time-dependent speciation and extinction rates, we derive an explicit formula for the distribution of the MRCA age of a uniformly sampled pair of extant lineages. This yields integral expressions, at any time, for both the variance of the empirical mean and the expected empirical variance. In the constant-rate birth-death case, we further obtain closed-form expressions for the expected empirical variance and describe its asymptotic behavior in the supercritical, critical and subcritical regimes.