ASYMPTOTIC CONDITIONAL DISTRIBUTIONS RELATED TO ONE-DIMENSIONAL GENERALIZED DIFFUSION PROCESSES

For a one-dimensional generalized diffusion process {X(t) : t >= 0} an interval I, we consider an expectation conditional on no hitting the end points of I. If the end points are not accessible, we take two sequences {xi(n)} and {eta(n)} which converge to the end points as n -> infinity, instead of end points. We obtain the asymptotic behavior of this conditional expectation as t -> infinity and n -> infinity. As an application of our results, we discuss the asymptotic conditional distribution and related quantities in population genetics.