# Coalescent theory - independence of coalescent times

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
• ...

source

• [Remi, you always make me read up things with your questions. At least half an hour is gone in this. But it's good :)].. As per wikipedia article on coalescent theory, population is also assumed to be very large for continuous approximation of coalescence time (exponential dist). – WYSIWYG May 6 '14 at 6:56
• @WYSIWYG ha ha I'm glad, I can help you seeking for more knowledge with my questions. I may try to write a longer background for my following questions, so to spare some time to the readers. – Remi.b May 6 '14 at 8:10
• @WYSIWYG Remi.b I like the reading for your questions, some have even prompted me to buy new books! – rg255 May 6 '14 at 9:27

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).

• +1 Good guy @daniel! You're answering to many of my questions. I am actually reading (a bit slowly because I have quite a lot of things to do beside this reading) John Wakeley's book, Coalescent Theory, an Introduction. I was expecting that the only assumptions for the independency of $T_i$ were those of the underlying Wright-Fisher or Moran model. I wasn't sure though. Thank you – Remi.b May 11 '14 at 8:39
• @Remi.b: It is good for me to work through some of these questions, which are interesting and thoughtful. I too am busy or I would work on more of them! Thanks. – daniel May 11 '14 at 10:41
• Your link to the Wakely paper is broke. Could you please write out the full reference so that it can be searched even if the link is broke? Thank you. – Hans Aug 15 at 23:33
• @Hans: Thank you I will get to this soon. Appreciate the heads-up. – daniel Aug 16 at 16:51