We consider a neutral dynamical model of biological diversity, where
individuals live and reproduce independently. They have i.i.d. lifetime
durations (which are not necessarily exponentially distributed) and give birth
(singly) at constant rate b. Such a genealogical tree is usually called a
splitting tree, and the population counting process (N_t;t\ge 0) is a
homogeneous, binary Crump--Mode--Jagers process. We assume that individuals
independently experience mutations at constant rate \theta during their
lifetimes, under the infinite-alleles assumption: each mutation instantaneously
confers a brand new type, called allele, to its carrier. We are interested in
the allele frequency spectrum at time t, i.e., the number A(t) of distinct
alleles represented in the population at time t, and more specifically, the
numbers A(k,t) of alleles represented by k individuals at time t,
k=1,2,...,N_t. We mainly use two classes of tools: coalescent point processes
and branching processes counted by random characteristics. We provide explicit
formulae for the expectation of A(k,t) in a coalescent point process
conditional on population size, which apply to the special case of splitting
trees. We separately derive the a.s. limits of A(k,t)/N_t and of A(t)/N_t
thanks to random characteristics. Last, we separately compute the expected
homozygosity by applying a method characterizing the dynamics of the tree
distribution as the origination time of the tree moves back in time, in the
spirit of backward Kolmogorov equations.Comment: 32 pages, 2 figures. Companion paper in preparation "Splitting trees
with neutral Poissonian mutations II: Large or old families
