You are currently browsing the tag archive for the ‘coming down from infinity’ tag.
If you haven’t read my previous post, I would suggest doing so before this. We denote by the partitions of . The thing to keep in mind here is that we want to think of a coalescent process as a history of lineage. Suppose we start with the trivial partition and think of each block as a member of some population. A coalescent process on is essentially defines ancestries, in the sense that if and belong to the same block of for some , then we think of that block as the common ancestor of and .
With this in mind, define the operator by
With some conditions, we can define the same operator on , the partitions of . So for example if and , then . The partition tells us in this case to merge the first and third block and leave the second block alone.