Spectral Density for Third Order Cumulants under Strong Mixing Conditions

If X≔{Xv: v∈Zd} is a strictly stationary random field, with X0 bounded and expressible as a sum of indicator functions satisfying certain conditions, if the mixing coefficient α(s) is summable over Zd (that, is, ∑m md−1α(m)<∞), and if a mixing condition involving three sets is satisfied, then the third order cumulant Cum(Xa, Xb, Xc) of X has a continuous spectral density. We do not begin with the assumption that the cumulants are absolutely summable.