Coalescent theory - independence of coalescent times

Question

Let $T_i$ be the time to coalesce from $n(t)=i+1$ to $n(t)=i$, where $n(t)$ is the number of sites that have not coalesced yet. In the below example the maximum $n(0)=6$.

As I understand it, many mathematical developments in coalescent theory depends on the fact that the random variables $T_i$ are independent (but not identically distributed). In other words...

$$f_{T_n, T_{n-1}, …. T_3, T_2}(t_n, t_{n-1},….,t_3, t_2) = \prod_{i=2}^n f_{T_i}(t_i)$$

What are the assumptions for this equation to hold true? Below are some suggestions

• No selection
• Selection is not varying through time
• Stable population size
• Random mating
• Both sexes have the same genetic background
• Both sexes have the same variance in fitness
• ...

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source

Answer 1

As long as members of a generation "randomly choose" their ancestor in the previous generation the law of independent probability (your equation) will hold.

Any study of coalescent theory begins with the Wright-Fisher model. The assumptions are:

• finite diploid population of constant size N,
• non-overlapping generations (simultaneous reproduction),
• random mating,
• no mutation, selection, or migration.

These assumptions are consistent with independent non-identically distributed waiting times. An example of an assumption under which independence no longer holds:

Individual B's random choice in generation 2 of ancestor A in generation 1 reduces the probability that individual C in generation 2 will choose A. In other words, the likelihood that A will give its genes to the next generation falls off with each new recipient. Then the independence no longer holds.

See, e.g., Deonier, Computational Genome Analysis (2005, Springer) at pp. 392 ff.

J. Wakely's paper Coalescent Theory: An Introduction (Systematic Biology, 58:1, Feb. 2009) may be one of the best overviews of this immense topic available. He mentions Kingman's 1982 mathematical proof (which I haven't gone through) of the coalescent process (Stochastic Processes and their Applications 13 (1982)--available as a free download from ScienceDirect).