Unlike the reduced k-body density matrices, the k-body cumulants generated from them are extensive entities. We formulate a theory directly in terms of these cumulants, based on stationarity of the energy with respect to variations induced by k-body excitation operators, that are in generalized normal order - as defined recently -with respect to the exact wavefunction. In terms of this new normal ordering, cumulants arise directly on taking expectation values. The equations determining the cumulants contain explicitly connected terms only, which ensures the extensivity of the theory. Our method resembles somewhat that of contracted Schrodinger equations, but differs in essential aspects from it.