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Asymptotic conditional distributions related to one-dimensional generalized diffusion processes. (English) Zbl 1122.60068
Let \(L:=(d/dm)(d/ds)\) be a generalized diffusion operator on a fixed open interval, and \(\{X(t): t\geq 0, \mathbb{P}_x: x\in I_m\}\) be a generalized diffusion process with the generator \(L\), where \(I_m\) is the support of the speed measure \(dm\). The authors investigate the asymptotic behavior of the conditional expectations \[ \mathbb{E}_x(f(X(\tau t))\mid t<\sigma_{\inf I_m}\wedge \sigma_{\sup I_m}) \quad \text{and} \quad \mathbb{E}_x(f(X(\tau t))\mid t<\sigma_{\inf I_m}< \sigma_{\sup I_m}), \] where \(\sigma_a\) is the first hitting time of \(a\in I_m\). The paper closes with an application of the main results to population genetics models.

60J60 Diffusion processes
60F05 Central limit and other weak theorems
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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