Nearest Neighbors of Multivariate Runs

We investigate the joint distributions of the number of nearest neighbor contacts between different objects in the context of runs-related statistics in multiple object systems. These distributions reveal spatial or temporal relations between runs that could not be answered by traditional run statistics, where the spatial or temporal relations between runs are ignored. To obtain the distributions of nearest neighbor contacts, we generalized the generating function approach we developed previously for run statistics. Explicit distributions and moments of the distributions were obtained. These generating functions also lead directly to the asymptotic distributions based on the singularity perturbation theory. Two kinds of nearest neighbor contacts are discussed. For each case the distributions for conditional and unconditional models are derived. By considering the nearest neighbor contacts in the context of runs and scans distributions, our study adds a new dimension to the existing knowledge in the field, opening up opportunities for further research to explore its full implications and potential applications.