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.
