I am studying a population of individuals over many generations. Each individual can take on a single trait from a range of several traits. In each generation I calculate the the share of the most common trait in the population. The result looks something like this:
Now I would like to say something along the lines of "the population has converged to a single trait at time T". To do this I have been using this naive notion of "convergence" where I simply take the first time point where the share of the most common trait at the time exceeded 95% (With no claim that this "convergence" has anything to do with convergence in the mathematical or statistical sense). I do this because I know from evolutionary biology (under a certain set of assumptions), that once this happens the odds of a new invasive trait being introduced via mutation that could take over the population are essentially zero.
I have been told by my colleague that I could improve my claims by pointing to convergence in distribution (or possibly other types of stochastic convergence). However, I don't see how I could show that my variable converges to any specific distribution (and what that distribution would be)? For starters my "trials" are not independent: the share in time T depends on the share in time T-1. Is it valid to even consider distribution in convergence in this case?
