2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/131281Symmetries of the auto-cumulant function (the 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 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, used to estimate general kth-order spectra, is also derived through the developed theory.19 pages, 3 figuresStatistics Theory37M10text