Multivariate lag-windows and group representations

Symmetries of the auto-cumulant function (a generalization of the auto-covariance function) of a kth-order stationary time series are derived through a connection with the symmetric group of degree k. Using the theory of group representations, symmetries of the auto-cumulant function are demystified and lag-window functions are symmetrized to satisfy these symmetries. A generalized Gabr–Rao optimal kernel is also derived through the developed theory.