Abstract: An important problem in statistical methods for marked point processes (MPPs) is to evaluate the relationship between points and marks, which can be developed under either the concept of independence or the concept of separability. Although both have been used, the connection between these two concepts is still unclear in the literature. The present article provides a way to evaluate such a connection, concluding that the concept of independence and the concept of separability are equivalent if the Kolmogorov consistency condition is satisfied, but not otherwise. We also provide a testing method to assess first-order independence between points and marks, where first-order independence is concluded if the test statistic is insignificant and first-order dependence is concluded if the test statistic is significant. The performance of the testing method is evaluated under simulation and case studies.

Key words and phrases: Counting measure, independence, Janossy measure, Kolmogorov consistency condition, marked point processes, separability.