Large deviations of the Wright-Fisher Process

In the present paper we will be concerned with the Wright-Fisher process of mathematical genetics in one of its simplest forms: the univariate case with no selection and with either no mutation or only one-way mutation. Consider a large biological population consisting of a single species in which a particular gene appears in two possible forms (alleles) A and a, say. Assume that we are dealing with a one-sex population (as with some plants) and that we are in the haploid case (in which chromosomes occur singly). The Wright-Fisher process models the way in which the proportion of individuals carrying the A form of the gene changes from generation to generation, as the population reproduces itself ([1], [4]). To be specific, let us first consider the case where the proportion of the A allele changes only through the effect of random sampling, with no mutation or selection. If the population consists of N individuals i of whom are of type A and N–i of type a, then the state of the process is the proportion $$ y = \frac{i}{N} $$ To produce an offspring generation from this population of “genes” we sample N times with replacement from it, thus keeping the size of the population constant. The probability that the proportion of A-alleles will make a transition from state $$ y = \frac{i}{N} $$ in the “current” generation to state $$ \tilde y = \frac{j} {N} $$ say, in the following generation is then 1.1$$ P\left( {y,\tilde y} \right) = \left( {\begin{array}{*{20}{c}} N \\ j \end{array}} \right){y^j}{\left( {1 - y} \right)^{N - j}} $$