A remark on a Fourier bounding method of proof for convergence of sums of periodograms (Q1388170)

One problem in time series analysis is studying the convergence of sums of periodograms. In dimension 1, \textit{D. R. Brillinger} [Time series. Data analysis and theory. Expand. ed. (1981; Zbl 0486.62095)] proved convergence using Fourier properties and associated asymptotics of the cumulants of finite Fourier transforms of the data series. In the present paper it is argued that this approach does not carry over to a lattice random field, this is, dimension 2. For this case, \textit{M. Rostenblatt}'s [Stationary sequences and random fields. (1985; Zbl 0597.62095)] method of proof is shown to be adequate. In order to have asymptotically correct confidence intervals, one needs to center these sums properly in the random field case.