ABSTRACT
This chapter deals with the elementary theory of cumulants in the multivariate case as well as the univariate case. Little prior knowledge is assumed other than some familiarity with moments and the notion of mathematical expectation, at least in the univariate case. Cumulants are normally introduced as functions of the moments. It is entirely natural to inquire at the outset why it is preferable to work with cumulants rather than moments since the two are entirely equivalent. The major departure from standard statistical notation is that we have chosen to use the same letter for both moments and cumulants. R. A. Fisher, in an astonishing tour de force, rediscovered cumulants, recognized their superiority over moments, developed the corresponding sample cumulants and cross-cumulants, gave the formulae for the cumulants and cross-cumulants of the sample cumulants and formulated combinatorial rules for computing the cumulants of such statistics.
