Scaling limits for the block counting process and the fixation line of a class of $\mathrm{\Lambda }$-coalescents

We provide scaling limits for the block counting process and the fixation line of $\mathrm{\Lambda }$-coalescents as the initial state $n$ tends to infinity under the assumption that the measure $\mathrm{\Lambda }$ on $[0,1]$ satisfies ${\int }_{[0,1]}{u}^{-1}(\mathrm{\Lambda }-b\lambda )(\mathrm{d}u)<\mathrm{\infty }$ for some $b>0$. Here $\lambda$ 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.