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.

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

Domaines: Probabilités [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⟩