3 problems

Let $\hat \ell_n:=\ell_n-\bar \ell_n$ be the total internal branch length of an $n$-coalescent, where $\ell_n$ is its total length and $\bar \ell_n$ is its total external branch le…

Let $\Xi\in\mathcal{X}$ be the finite measure governing a general $\Xi$-coalescent, and let $\psi^\Xi_n$ denote the quantity controlling the additional sample-size requirement for…

Let $\Lambda$ be a finite measure on $[0,1]$ with $\Lambda(\{1\})=0$, and let $\psi(q)=\int_{[0,1]}(e^{-qx}-1+qx)x^{-2}\Lambda(dx)$. For a sample of size $n$, let $X_n$ denote eith…