Traces and Cumulants of Quadratic Forms in Normal Variables

Summary The cumulants of a quadratic form in n independent standardized variables are identified with the traces of the powers of the matrix of the form. This identification leads to some very simple proofs of well-known theorems. It is proved that only in a normal system is it possible to have non-trivial linear transformations from one set of independent variables to another set of independent variables. Matrix proofs are given for the Cochran theorems paying special attention to the traces of the matrices involved. A method of determining the multivariate moments and cumulants of any linear combination of the usual Wishart variables is suggested.