Extension of the matrix bartlett´s formula to the third order and to noisy models

This paper focuses on the extension of the asymptotic covariance of the sample covariance (denoted Bartlett´s formula) of linear processes to third-order sample cumulant and to noisy linear processes. Thanks to a matrix polyspectral representation, closed-form expressions of the asymptotic covariance and cross covariance of the sample second and third moments are derived in a straightforward manner. As an application of these extended formulae, we enhance the sensitivity of the asymptotic performance of estimated ARMA parameters by an arbitrary third order-based algorithm to the spectrum of colored additive noise.