On hitting probabilities of beta coalescents and absorption times of coalescents that come down from infinity

Let X = (Xk)k=0,1,... denote the jump chain of the block counting process of the Λ-coalescent with Λ = β(2 − α, α) being the beta distribution with parameter α ∈ (0, 2). A solution for the hitting probability h(n,m) that the chain X ever visits the state m, conditional that it starts in the state X0 = n, is obtained via an analytic method based on generating functions. For α ∈ (1, 2) the results are applied to characterize the distribution of the almost sure limit τ of the absorption times τn of the coalescent restricted to a sample of size n. The latter result is generalized to arbitrary exchangeable coalescents (Ξ-coalescents) that come down from infinity. The results generalize those obtained for the particular case α = 1 in Möhle, M. (2014) Asymptotic hitting probabilities for the Bolthausen–Sznitman coalescent, J. Appl. Probab. 51A, to appear. This article furthermore supplements the work of Hénard, O. (2013), The fixation line, Preprint, arXiv:1307.0784.