Title: Asymptotic Goodness-of-Fit Tests for Point Processes Based on Scaled Empirical K-Functions,
Abstract: We study sequences of scaled edge-corrected empirical (generalized)
K-functions (modifying Ripley's K-function) each of them constructed from a
single observation of a $d$-dimensional fourth-order stationary point process
in a sampling window W_n which grows together with some scaling rate
unboundedly as n --> infty. Under some natural assumptions it is shown that the
normalized difference between scaled empirical and scaled theoretical
K-function converges weakly to a mean zero Gaussian process with simple
covariance function. This result suggests discrepancy measures between
empirical and theoretical K-function with known limit distribution which allow
to perform goodness-of-fit tests for checking a hypothesized point process
based only on its intensity and (generalized) K-function. Similar test
statistics are derived for testing the hypothesis that two independent point
processes in W_n have the same distribution without explicit knowledge of their
intensities and K-functions.