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Asymptotic Theory for Estimators of High-Order Statistics of Stationary Processes

High-order cumulants and high-order spectra play an important role in the theory of stationary processes. For nonlinear processes, it is quite challenging to develop an asymptotic theory for their estimators. This paper presents a systematic asymptotic theory for estimators of high-order moments for a general class of stationary processes, using the framework of functional dependence measures. In particular, we prove the asymptotic normality of the estimators and establish a uniform convergence rate. We also provide a sufficient condition for the summability of high-order cumulants. Based on the latter, we prove consistency and asymptotic normality of the third-order spectra or bispectrum estimators under mild and easily verifiable conditions.