CONVERGENCE RATE OF THE LIMIT THEOREM OF A GALTON-WATSON TREE WITH NEUTRAL MUTATIONS

Xinxin Chen (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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 dierence between the rescaled size of sub-families and corresponding number of mutations. We nd out that it converges in law to some normal distribution with mean zero and whose variance is associated with the CSBP.

Keywords: Levy-Ito decomposition; Weak convergence; Donsker's invariance principle; Branching process; Donsker's invariance principle.; Neutral mutations

Domains: Probability [math.PR]

Xinxin Chen. CONVERGENCE RATE OF THE LIMIT THEOREM OF A GALTON-WATSON TREE WITH NEUTRAL MUTATIONS. Statistics and Probability Letters, 2013, 83 (2), pp.588-595. ⟨hal-00569369⟩