A CLT for the periodograms of a ϱ∗-mixing random field

If X is a centered, ϱ∗-mixing random field with certain weak stationarity conditions and finite fourth moments, then the fourth moments of block sums Sn grow as the square of block size n. Further, n−2d∥Sn∥44 converges to a limit which can be expressed in terms of the spectral density. This result is used to derive a CLT for periodograms of the spectral density of strictly stationary X.