Asymptotic Normality of Bispectral Estimates

Abstract

The work presented here is a continuation of that presented in an earlier paper, M. Rosenblatt and J. Van Ness [15], in which the basic properties (unbiasedness and consistency) of certain estimates of the bispectrum and bispectral density are discussed. (The bispectrum can be thought of as the Fourier transform of the third-order moment function of the process.) The present paper is concerned with the asymptotic distribution of these estimates. One would expect that under certain regularity conditions these estimates would have distributions which tend to complex normal distributions. The following develops two different sets of such conditions either of which suffice. The first set involves a uniform summability condition on the first six cumulants of a sequence of processes obtained from the original process by projecting it onto a certain sequence of Borel fields. The second, and much more intuitively meaningful, set involves the strong mixing condition (see Rosenblatt [11], [12]; Kolmogorov and Rozanov [6], and Volkonskii and Rozanov [18] e.g.). The calculations in the earlier paper were carried out for the continuous parameter case; here we restrict ourselves to the discrete parameter case. For reasons for interest in polyspectra see, for example, Rosenblatt and Van Ness [15]; Hasselmann, Munk and MacDonald [4]; and Brillinger [2].