Splitting trees with neutral Poissonian mutations II: Largest and oldest families

We consider a supercritical branching population, where individuals have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate. We assume that individuals independently experience neutral mutations, at constant rate θ during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called allele or haplotype, to its carrier. The type carried by a mother at the time when she gives birth is transmitted to the newborn. interested in the sizes and ages at time t of the clonal families carrying the most abundant alleles or the oldest ones, as t → ∞ , on the survival event. Intuitively, the results must depend on how the mutation rate θ and the Malthusian parameter α > 0 compare. Hereafter, N ≡ N t is the population size at time t , constants a , c are scaling constants, whereas k , k ′ are explicit positive constants which depend on the parameters of the model. > θ , the most abundant families are also the oldest ones, they have size c N 1 − θ / α and age t − a . < θ , the oldest families have age ( α / θ ) t + a and tight sizes; the most abundant families have sizes k log ( N ) − k ′ log log ( N ) + c and all have age ( θ − α ) − 1 log ( t ) . = θ , the oldest families have age k t − k ′ log ( t ) + a and tight sizes; the most abundant families have sizes ( k log ( N ) − k ′ log log ( N ) + c ) 2 and all have age t / 2 . informal results can be stated rigorously in expectation. Relying heavily on the theory of coalescent point processes (Popovic, 2004, Lambert, 2010), we are also able, when α ≤ θ , to show convergence in distribution of the joint, properly scaled ages and sizes of the most abundant/oldest families and to specify the limits as some explicit Cox processes. s in deep contrast with the largest/oldest families in the standard coalescent with Poissonian mutations, which converge to some point processes after being rescaled by N (Donnelly and Tavaré 1986, Ewens, 2005, Durrett, 2008).