This paper focuses on the extension of the asymptotic covariance of the sample covariance (denoted Bartlett's formula) of linear processes to third- and fourth-order sample cumulant and to noisy linear processes. Closed-form expressions of the asymptotic covariance and cross-covariance of the sample second-, third- and fourth-order cumulants are derived in a relatively straightforward manner thanks to a matrix polyspectral representation and a symbolic calculus akin to a high level language. As an application of these extended formulae, we underscore the sensitivity of the asymptotic performance of estimated ARMA parameters by an arbitrary third- or fourth order-based algorithm with respect to the signal to noise ratio, the spectra of the linear process, and the colored additive noise.