ABSTRACT
To illustrate the result of Eq. (2.12) consider the evaluation of cum{X( 1)0X(2l,x(Jl}, where the random matrices X(l),l= 1,2 and 3 are all of dimensions d x d. The original partition is { 12!3} and the indecomposable partitions are {123}, {1123} and {1312}. If Idxd is ad x d identity matrix and htxc/) is the d 2 x d2 permuted identity matrix such that I(ddJ{X( 3l 0 X(2l}J(ci,ci) = X(2) 0 X(3l, then from Eq. (2.12), we get
cum{X(IJ 0 X(2l, X( 3)} = cum{X( 1l, X(2), X(JJ} + cum{X( 1)} 0 cum{X(2J ,X(3l}
We now use the above results to derive expressions for the kth order cumulants of d dimensional stationary time series { X1}.
