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The Mittag-Leffler process and a scaling limit for the block counting process of the Bolthausen-Sznitman coalescent
The Mittag–Leffler process X = (Xt)t≥0 is introduced. This Markov process has the property that its marginal random variables Xt are Mittag–Leffler distributed with parameter e, t ∈ [0,∞), and the…
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On the number of collisions in beta(2, b)-coalescents
Expansions are provided for the moments of the number of collisions $X_n$ in the $\beta(2,b)$-coalescent restricted to the set $\{1,...,n\}$. We verify that $X_n/\mathbb{E}X_n$ converges almost…
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On the block counting process and the fixation line of the Bolthausen-Sznitman coalescent
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On a random recursion related to absorption times of death Markov chains
Let $X_1,X_2,...$ be a sequence of random variables satisfying the distributional recursion $X_1=0$ and $X_n= X_{n-I_n}+1$ for $n=2,3,...$, where $I_n$ is a random variable with values in…
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On the block counting process and the fixation line of exchangeable coalescents
We study the block counting process and the fixation line of exchangeable coalescents. Formulas for the infinitesimal rates of both processes are provided. It is shown that the block counting process…
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Asymptotics of continuous-time discrete state space branching processes for large initial state
Scaling limits for continuous-time branching processes with discrete state space are provided as the initial state tends to infinity. Depending on the finiteness or non-finiteness of the mean and/or…
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Absorption time and tree length of the Kingman coalescent and the Gumbel distribution
Formulas are provided for the cumulants and the moments of the time $T$ back to the most recent common ancestor of the Kingman coalescent. It is shown that both the $j$th cumulant and the $j$th…
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On the stationary distribution of the block counting process for population models with mutation and selection
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The collision spectrum of $\Lambda$-coalescents
$\Lambda$-coalescents model the evolution of a coalescing system in which any number of blocks randomly sampled from the whole may merge into a larger block. For the coalescent restricted to…
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On the size of the block of 1 for $\varXi$-coalescents with dust
We study the frequency process $f_1$ of the block of 1 for a $\varXi$-coalescent $\varPi$ with dust. If $\varPi$ stays infinite, $f_1$ is a jump-hold process which can be expressed as a sum of broken…
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