2021/07/14 by Martin Möhle, Möhle, Martin, Benedict Vetter +1
Mathematics · #60J90 (Primary) 60J27 (Secondary) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2107.06718
openalex publication_date 2021/07/14 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
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 ∫[0,1]u-1(Λ-bλ)(\rm du)0. Here λ denotes the Lebesgue measure. The main result states that the block counting process, properly logarithmically scaled, converges in the Skorohod space to an Ornstein--Uhlenbeck type 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 Lambda and then prove the uniform convergence of both parts separately.