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- Title
Scaling limits for the block counting process and the fixation line for a class of Λ-coalescents.
- Authors
Möhle, Martin; Vetter, Benedict
- 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 ∫[0,1] u-1|Λ - bλ|(du) < ∞ 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.
- Subjects
COUNTING; LEBESGUE measure; STOCHASTIC convergence; PARAMETERS (Statistics); INFINITY (Mathematics)
- Publication
ALEA. Latin American Journal of Probability & Mathematical Statistics, 2022, Vol 19, Issue 1, p641
- ISSN
1980-0436
- Publication type
Article
- DOI
10.30757/ALEA.v19-25