AI 中文总结
研究实直线上非退化平稳中心高斯过程相继零点最小间距，在协方差核及其导数特定条件下，经重标度最小间距收敛到泊松点过程，其出现位置趋于均匀分布，可推导第k个最小间距极限密度。
AI 中文摘要
本文研究实直线上非退化、光滑、平稳、中心高斯过程相继零点之间的最小间距，假设协方差核κ(x)及其导数在|x|→∞时趋于0。我们证明，经过重标度后，最小间距收敛到具有特定速率的泊松点过程。此外，这些最小间距出现的位置趋于均匀分布。因此，我们可以推导出第k个最小间距的极限密度。
英文摘要
In this paper, we study the smallest gaps between successive zeros of nondegenerate smooth stationary centered Gaussian processes on the real line with the assumption that the covariance kernel $κ(x)$ and its derivatives decay to 0 as $|x|\to\infty$. We prove that, after rescaling, the smallest gaps converge to a Poisson point process with a specific rate. Moreover, the positions where these smallest gaps occur tend to a uniform distribution. Consequently, we can derive the limiting density for the $k$-th smallest gap.
CommentsCompared with the published version, in this version we have added some recent results at the end of the introduction, and included the corresponding reference [5]
Journal refJournal of Functional Analysis, Volume 287, Issue 4, 100493 (2024)
