Asymptotic conditional distributions related to one-dimensional generalized diffusion processes

Abstract

For a one-dimensional generalized diffusion process $\{X(t) : t \geq O\}$ 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 $\{ \xi_{n}\}$ and $\{ \eta_{n} \}$ which converge to the end points as $n \rightarrow \infty$, instead of end points. We obtain the asymptotic behavior of this conditional expectation as $t \rightarrow \infty$ and $n \rightarrow \infty$. As an application of our results, we discuss the asymptotic conditional distribution and related quantities in population genetics.