Bartlett's formulae?Closed forms and recurrent equations

Abstract

We show that the entries of the asymptotic covariance matrix of the sample autocovariances and autocorrelations of a stationary process can be expressed in terms of the square of its spectral density. This leads to closed form expressions and fast computational algorithms.

A robust whiteness test for the identification of discrete-time linear models: Use of orthonormal transfer functions

“…Proof of 2. It is known (see for example [2]) that the covariance associated to the output of the ARMA process 1 (1−|p0|z −1 ) 2 driven by a white noise with variance equal to 1 is 1+|p0| 2…”

On the asymptotic properties of multivariate sample autocovariances

“…In the multivariate case the infinite sums in Bartlett's formulas can be replaced by the autocovariances corresponding to the tensor square of the spectral density matrix of the process (see [3]). For univariate processes the latter is simply the square of the spectral density (see [2,4,10]). …”