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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.

MSC:
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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