The asymptotic distribution of the Whittaker periodogram and a related chi-squared statistic for stationary processes

Abstract

We find the asymptotic distribution of two statistics that have been employed in the search for periodicities. Both statistics are based on Buys-Ballot tables. The first, the Whittaker periodogram, is computed for ordinary time series. The second, a X 2 statistic, is computed for point processes. These statistics have been viewed as competitors of power spectrum estimates. We find that when autocorrelation is present in the process involved, the large-sample distributions of the statistics differ from the null distributions used in testing. The statistics therefore are appropriate only when the null hypothesis is that the process involved has a constant power spectrum.