  Factorial moments and cumulants are usually defined with respect to the
unconditioned Poisson process. Conditioning a sample by selecting events of a
given overall multiplicity $N$ necessarily introduces correlations. By means of
Edgeworth expansions, we derive generalized cumulants which define correlations
with respect to an arbitrary process rather than just the Poisson case. The
results are applied to correlation measurements at fixed $N$, to redefining
short-range vs.\ long-range correlations and to normalization issues.
