  For general thinning procedures, its inverse operation, the condensing, is
studied and a link to integration-by-parts formulas is established. This
extends the recent results on that link for independent thinnings of point
processes to general thinnings of finite point processes. In particular, the
classical integration-by-parts formulas appear as the example of independent
thinnings. Moreover, the representation of the splitting kernel of finite point
processes in terms of its reduced Palm kernels is extended to general
thinnings.
  This link is studied in the context of discrete random variables and yields
analogue characterizations of their distributions. Results on independent
thinnings are complemented by a discrete stick breaking characterization of
distributions.
