The ranked mass process associated with a Lambda-coalescent

Question

I am reading a paper by Pitman (1999), and I am confused by his Corollary 7. First some notation so that I can explain my confusion:

• $\mathcal{P}_\infty$ is the space of partitions of $\mathbb{N}$, $\mathcal{P}_n$ is its restriction to $[n] = \{1, 2, \ldots, n\}$.
• $\Pi_\infty^\pi(t)$ is a $\Lambda$-coalescent on $\mathcal{P}_\infty$ with $\Pi_\infty^\pi(0) = \pi \in \mathcal{P}_\infty$ and $\Pi_n^\pi(t)$ is its restriction to $\mathcal{P}_n$. A standard $\Lambda$-coalescent is one that starts at $\pi = \{\{1\},\{2\},\ldots\}$.
• $\mathcal{S}^\downarrow = \{x \in \ell^1: x_1 \geq x_2 \geq \ldots, \, \sum x_i = 1\}$. In other words, $\mathcal{S}^{\downarrow}$ is the set of ranked probability distributions on $\mathbb{N}$.

Below is Corollary 7.

I am confused about the second sentece in the corollary. Clearly, there is a ranked rearrangement of the masses of $\pi$, but what is meant by "The $\mathbf{x}$-masses of $\pi$"? What I understand is that $\mathbf{x}$ is a particular element of $\mathcal{S}^\downarrow$ and $\pi$ is a particular element of $\mathcal{P}_\infty$ whose ranked mass rearrangement may or may not coincide with $\mathbf{x}$, hence my confusion.

Could someone explain to me what is meant by the $\mathbf{x}$-masses of $\pi$ and what the process $((\mathbf{x}, \Pi_\infty(t)), t\geq 0)$ looks like?

Answer 1

$\newcommand{\bx}{{\mathbf{x}}}$

The idea is that the vector $\bx = (x_i, i\in \mathbb{N})$ assigns a mass to every point of $\mathbb{N}$ (with the masses in decreasing order and summing to $1$).

Now if $B$ is a block, i.e. a subset of $\mathbb{N}$, its "$\bx$-mass" is $\sum_{i\in B} x_i$, the sum of the masses of all the points contained in the block, where "mass" is defined by the vector $\bx$.

Then for a partition of $\mathbb{N}$, say $\pi=(B_1, B_2, \dots)\in \mathcal{P}_\infty$, you can consider the vector of the $\bx$-masses of its blocks, $(\sum_{i\in B_1} x_i, \sum_{i\in B_2} x_i, \dots)$. These $\bx$-masses sum to $1$ and you can put them into decreasing order. This defines $(\bx, \pi)$.

A special case is that if $\pi$ is the partition of $\mathbb{N}$ into singletons, then $(\bx, \pi)$ is simply $\bx$.

So if you have a coalescent $\Pi(t)$ taking values in partitions of $\mathbb{N}$, whose initial state is the partition into singletons, then the process $(\bx, \Pi(t))$ is a coalescent in the sense of partitions of unit mass, with initial state $\bx$.