Generalized moments and cumulants for samples of fixed multiplicity

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.