The nearest neighbor statistics for X-ray source counts I.The method
  Most of the X-ray background (XRB) is generated by discrete X-ray sources. It is likely that still unresolved fraction of the XRB is composed from a population of the weak sources below the present detection thresholds and a truly diffuse component. It is a matter of discussion a nature of these weak sources. The goal of the present paper is to explore the effectiveness of the nearest neighbor statistics (NNST) of the photon distribution for the investigation of the number counts of the very weak sources. All the sources generating at least two counts each induce a nonrandom distribution of counts. This distribution is analyzed by means of the NNST. Using the basic probability equations, the relationships between the source number counts N(S) and the NNST are derived. It is shown that the method yields constraints on the N(S) relationship below the regular discrete source detection threshold. The NNST was applied to the medium deep Chandra pointing to assess the source counts N(S) at flux levels attainable only with the very deep exposures. The results are in good agreement with the direct source counts based in the Chandra Deep Fields (CDF). In the next paper of this series the NNST will be applied to the the CDF to assess the source counts below the present flux limits. 
Andrzej M. So{\l}tan
astro-ph.CO