The $K$-function is arguably the most important functional summary statistic for spatial point processes. It is used extensively for goodness-of-fit testing and in connection with minimum contrast estimation for parametric spatial point process models. It is thus pertinent to understand the asymptotic properties of estimates of the $K$-function. In this paper we derive the functional asymptotic distribution for the $K$-function estimator. Contrary to previous papers on functional convergence we consider the case of an inhomogeneous intensity function. We moreover handle the fact that practical $K$-function estimators rely on plugging in an estimate of the intensity function. This removes two serious limitations of the existing literature.

翻译：K-函数可以说是空间点过程最重要的泛函汇总统计量。它被广泛用于拟合优度检验以及与参数空间点过程模型的最小对比估计相关的应用。因此，理解K-函数估计量的渐近性质具有重要意义。本文推导了K-函数估计量的泛函渐近分布。与以往研究泛函收敛性的论文不同，我们考虑了非齐次强度函数的情形。此外，我们处理了实际应用中K-函数估计量需代入强度函数估计值这一事实。这消除了现有文献中两个严重的局限性。