Symmetry properties of higher spectral densities and nonparametric estimates of them

The symmetry properties of the spectral densitiesf ^(v)() of order v, equal to 4, 5, and 6 of a stationary stochastic process {(k), k Z} are studied. It is established that the fonctionf ^(v)(), R^v–1 is completely determined by its values on a set whose (v– 1)-dimensional volume is ⁴/6, ⁵/15, and ⁵/45 when v is equal to 4, 5, and 6, respectively. Results on the statistical estimation of the spectral densities of orders v equal to 3 and 4 with respect to samples of finite size are also given.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 1–4, 1988.