Asymptotic Moments and Distribution of Nearest Marker Distances When Major Genes and Marker Polymorphisms are Unevenly Distributed

The asymptotic mean and variance of the distance from a major gene to the nearest marker are derived for the case when the markers are distributed with density m(x) along the chromosome and the major gene with density g(x). The asymptotic distribution was derived by analogy with Poisson processes. For a given g(x), the density of markers that minimised the expected distance to the nearest marker was proportional to root g(x); expected squared distance was minimised when the density of markers was proportional to root g(x). Comparison with simulations showed the asymptotic moments and distribution to be accurate when the number of markers n greater than or equal to 100, but even for n greater than or equal to 20 the first moment was considerably better than that derived from assuming uniformly distributed marker loci.