Scaling limits for the block counting process and the fixation line for a class of Λ-coalescents

Abstract

. We provide scaling limits for the block counting process and the fixation line of Λ coalescents as the initial state n tends to infinity under the assumption that the measure Λ on [0 , 1] satisfies (cid:82) [0 , 1] u − 1 | Λ − bλ | (d u ) < ∞ for some b ≥ 0 . Here λ denotes the Lebesgue measure on [0 , 1] . The main result states that the block counting process, properly transformed, converges in the Skorohod space to a generalized Ornstein–Uhlenbeck process as n tends to infinity. The result is applied to beta coalescents with parameters 1 and b > 0 . We split the generators into two parts by additively decomposing Λ into a ‘Bolthausen–Sznitman part’ bλ and a ‘dust part’ Λ − bλ and then prove the uniform convergence of both parts separately.