Kingman's coalescent

A partition-valued Markov chain in which every unordered pair of current blocks merges at rate one. On $n$ initial singleton blocks, the holding time with $j$ blocks is exponential with rate $\left(\genfrac{}{}{0ex}{}{j}{2}\right)$. Restrictions to subsets of labels have the same law.

In Kingman's coalescent, the epoch with $j$ ancestral lineages lasts $T_j\sim \mathrm{Exp}((\genfrac{}{}{0ex}{}{j}{2}))$, independently across epochs. If each lineage carries an independent Poisson process of mutations with rate $\theta /2$, its epoch mutation count satisfies $Y_j\mid T_j\sim \mathrm{Poisson}(j\theta T_j/2)$. The total mutation count is $S={\sum }_{j=2}^n{Y}_j$. Under the infinite sites mutation model, this also counts segregating sites; without that assumption it still counts mutation events, which need not correspond one-to-one to observed polymorphisms.

The independent mutation counts in coalescent epochs have the geometric mutation count in a coalescent epoch laws. Summing their expectations and variances gives the displayed formulas, where $H_m={\sum }_{i=1}^{m}1/i$ and $H_m^{(2)}={\sum }_{i=1}^{m}1/i^2$. The extra quadratic term records randomness of the ancestral tree, beyond the conditional Poisson distribution variance.

Integrating the Poisson distribution probability generating function over the exponential distribution of the epoch length givesThus $Y_j$ has a geometric distribution on $\{0,1,\dots \}$, with expectation $\theta /(j-1)$ and variance $\theta /(j-1)+{\theta }^2/(j-1{)}^2$. The support convention matters: the count includes zero mutations.

The epoch lengths in Kingman's coalescent are independent random variables. Conditional on these lengths, mutation counts on disjoint tree pieces are independent Poisson random variables. Their conditional laws depend only on the corresponding epoch length, so integrating the product of conditional laws against the independent epoch laws leaves independent unconditional epoch mutation counts. This justifies adding their variances, rather than assuming independence from conditional independence alone.

In coalescent units, independent waiting times satisfy $T_j\sim \mathrm{Exp}(j(j-1)/2)$, and total branch length is $L={\sum }_{j=2}^{n}j{T}_j$. Therefore $EL=2{\sum }_{i=1}^{n-1}1/i$. The variables $j{T}_j/2$ are independent exponentials with rates $j-1$, so $W=L/2$ has the same distribution as the maximum of $n-1$ independent unit exponentials, with density $(n-1)e^{-w}(1-e^{-w}{)}^{n-2}$. The last identity follows either from its Laplace transform or from the independent spacings of exponential order statistics.

The consistent partition process on $\mathbb{N}$ whose finite restrictions are Kingman's coalescent. It starts with infinitely many singleton blocks.

For $t>0$, the number $B_t$ of blocks is finite almost surely. On a finite restriction, the expected time to reach $m$ blocks is at most ${\sum }_{j>m}2/[j(j-1)]=2/m$. Markov inequality, followed by increasing the restriction, gives $\mathbb{P}(B_t>m)\le 2/(mt)$. Let $m\to \infty$, then use rational times and monotonicity to obtain the assertion simultaneously for every $t>0$.