In spite of mutations that – given the state of a population – occur independently in each individual, members of (especially: small) populations exhibit striking similarities. This is due to genetic drift, one of the most important forces of population genetics. Simply put, the reason for this phenomenon is that in a population, on the one hand, new variants are introduced randomly by (neutral) mutations, and, on the other, many variants are also randomly lost as not all members of the current generation pass their genetic material to the next one.

This situation is clearly presented in the following model of Wright and Fisher [49, 134, 141]. We suppose the population in question to be composed of 2N individuals; in doing so we identify individuals with chromosomes (that come in pairs), or even with corresponding loci on these chromosomes. We assume there are only two possible alleles (variants) at this locus: A and a. The size of the population is kept constant all the time, and we consider its evolution in discrete non-overlapping generations formed as follows: an individual in the daughter generation is the same as its parent (reproduction is asexual) and the parent is assumed to be chosen from the parent generation randomly, with all parents being equally probable. In other words, the daughter generation is formed by 2N independent draws with replacement from the parent generation. In each draw all parents are equally likely to be chosen and daughters have the same allele as their parents. It should be noted here that such sampling procedure models the genetic drift by allowing some parents not to be selected for reproduction, and hence not contributing to the genetic pool.

Then, the state of the population at time n ≥ 0 is conveniently described by a single random variable Xn with values in {0, …, 2N} being equal to the number of individuals of type A. The sequence Xn, n ≥ 0 is a time-homogeneous

Markov chain with transition probabilities:

In other words, if Xn = k, then Xn+1 is a binomial random variable with parameter.