We study the frequency process

Independently introduced in [

The family of

Moreover, in [

For two particular coalescents without dust, further properties of

Different specific aspects of the block of 1 have been analysed for different

External branch length: The waiting time for the first jump of the block of 1 in the

Minimal clade size: The size

The number of blocks involved in the first merger of the block of 1, see [

The number of blocks involved in the last merger of the block of 1, see [

The small-time behaviour of the block of 1, see [

Dirac coalescents (

We further characterise

Our key motivation was to provide a more detailed description of the jump chain of

Compared to the known results listed above for the minimal clade size for dust-free coalescents, the minimal clade size is much larger asymptotically for

The law of

Our main tool for the proofs is Schweinsberg’s Poisson construction of the

We recall the construction of a

Let

The block of 1 can only merge at Poisson points

Now, we project the points of

For a

When constructing simple

Since many proofs will build on the properties of different sets of exchangeable indicators, we collect some well-known properties in the following

These properties essentially follow from the de Finetti representation of an infinite series of exchangeable variables as conditionally i.i.d. variables. The lemma is a collection of well-known properties as e.g. described in [

An infinite exchangeable sequence is conditionally i.i.d. given an almost surely unique random measure

A crucial assumption for our results is that the

We recall the condition for

Let

First, assume

Now assume

Finally, assume

As in Lemma

The value of

We consider the jump chain of

By the Poisson construction the block of 1 for

Let

To describe

We consider

Now, consider the asymptotic frequency

Applied successively, this shows that the newly formed block at the

If

We have

Assume

The property (i) of

If

To prove Proposition

We adjust the proof of [

From Eq. (

Since

Since

In order to verify that the jump chain

We can express all events in terms of the independent waiting times for Poisson points, i.e. the successive differences between the first component

The event

Taking the ratio shows that

We provide a concrete example showing that the random variables

Choose

Jason Schweinsberg and an anonymous reviewer pointed out that

F. Freund was funded by the grant FR 3633/2-1 of the German Research Foundation (DFG) within the priority program 1590 “Probabilistic Structures in Evolution”.