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Möhle, Martin; Vetter, Benedict

Scaling limits for the block counting process and the fixation line for a class of \(\Lambda\)-coalescents. (English) Zbl 1494.60089

ALEA, Lat. Am. J. Probab. Math. Stat. 19, No. 1, 641-664 (2022).
Summary: We provide scaling limits for the block counting process and the fixation line of \(\Lambda\)-coalescents as the initial statentends to infinity under the assumption that the measure \(\Lambda\) on [0,1] satisfies \(\int_{[0,1]}u^{-1}|\Lambda-b\lambda|(\mathrm{d}u)<\infty\) for some \(b \geq 0\). Here \(\lambda\) 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 \(\Lambda\) into a ‘Bolthausen-Sznitman part’ \(b \lambda\) and a ‘dust part’ \(\Lambda-b \lambda\) and then prove the uniform convergence of both parts separately.

Cited in 2 Documents

MSC:

60J90 Coalescent processes
60J27 Continuous-time Markov processes on discrete state spaces

Keywords:

block counting process; Bolthausen-Sznitman coalescent; coalescent; dust; fixation line; generalized Ornstein-Uhlenbeck process; time-inhomogeneous process; weak convergence
PDFBibTeX XMLCite
\textit{M. Möhle} and \textit{B. Vetter}, ALEA, Lat. Am. J. Probab. Math. Stat. 19, No. 1, 641--664 (2022; Zbl 1494.60089)
Full Text: arXiv Link

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