Probabilistic multi-catalogue positional cross-match
  We lay the foundations of a statistical framework for multi-catalogue
cross-correlation and cross-identification based on explicit simplified
catalogue models. A proper identification process should rely on both
astrometric and photometric data. Under some conditions, the astrometric part
and the photometric part can be processed separately and merged a posteriori to
provide a single global probability of identification. The present paper
addresses almost exclusively the astrometrical part and specifies the proper
probabilities to be merged with photometric likelihoods.
  To select matching candidates in n catalogues, we used the Chi (or,
indifferently, the Chi-square) test with 2(n-1) degrees of freedom. We thus
call this cross-match a chi-match. In order to use Bayes' formula, we
considered exhaustive sets of hypotheses based on combinatorial analysis. The
volume of the Chi-test domain of acceptance -- a 2(n-1)-dimensional acceptance
ellipsoid -- is used to estimate the expected numbers of spurious associations.
We derived priors for those numbers using a frequentist approach relying on
simple geometrical considerations. Likelihoods are based on standard Rayleigh,
Chi and Poisson distributions that we normalized over the Chi-test acceptance
domain. We validated our theoretical results by generating and cross-matching
synthetic catalogues.
  The results we obtain do not depend on the order used to cross-correlate the
catalogues. We applied the formalism described in the present paper to build
the multi-wavelength catalogues used for the science cases of the ARCHES
(Astronomical Resource Cross-matching for High Energy Studies) project. Our
cross-matching engine is publicly available through a multi-purpose web
interface. In a longer term, we plan to integrate this tool into the CDS XMatch
Service.
