All-pass matrices, the positive real lemma, and unit canonical correlations between future and past
Stationary vector ARMA processes x(t), \(t=0\), \(\pm 1\), \(\pm 2,...\), that are full rank are considered. It is shown, with E(z) denoting the `symbol' of the canonical correlation operator, that the number of zeros (counting multiplicities) on \(| z| =1\) of the transfer function, V(z), from innovations to outputs \(=\) (change in arg det E(z) around \(| z| =1)/2\pi =the\) number of unit Hankel singular values of E(s). It follows that the number of unit canonical correlations between the future x(t), \(t>0\), and the past x(t), \(t\leq 0\), is equal to the number of zeros of V(z) on \(| z| =1\), which is the result of \textit{E. J. Hannan} and \textit{D. S. Poskitt} [Ann. Stat. 16, No.2, 784-790 (1988; Zbl 0646.62082)]. This note, however, proves this result directly from the properties of E(z), which Hannan and Poskitt suggested should be possible. It is also shown that the result can be obtained using an entirely state space approach via the positive real lemma.
scientific article; zbMATH DE number 4086825
scientific article