Convergence Rate of the Limit Theorem of a Galton-watson Tree with Neutral Mutations

We consider a Galton-Watson branching process with neutral mutations (infinite alleles model), and we decompose the entire population into sub-families of individuals carrying the same allele. Bertoin[3] has established the description of the asymptotic shape of the process of the sizes of the allelic sub-families when the initial population is large and the mutation rate small. The limit in law is a certain continuous state-space branching process (CSBP). Further, we consider the difference between the rescaled size of sub-families and corresponding number of mutations. We find out that it converges in law to some normal distribution with mean zero and whose variance is associated with the CSBP.