ASYMPTOTIC CONDITIONAL DISTRIBUTIONS RELATED TO ONE-DIMENSIONAL GENERALIZED DIFFUSION PROCESSES

For a one-dimensional generalized diffusion process {X(t): t ≥ 0} on 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 {ξn} and {ηn} which converge to the end points as n → ∞, instead of end points. We obtain the asymptotic behavior of this conditional expectation as t → ∞ and n → ∞. As an application of our results, we discuss the asymptotic conditional distribution and related quantities in population genetics.