Baxter's inequality for finite predictor coefficients of multivariate long-memory stationary processes (Q2405218)

Due to the large area of applicability of Baxter's inequality in various domains, a lot of specific extensions have been done. For extending it to the setup of multivariate stationary processes, the boundedness of the spectral density is essential. An extension where the boundedness is not essential was made in [the first two authors, Ann. Stat. 34, No. 2, 973--993 (2006; Zbl 1098.62120)] where Baxter's inequality was established for univariate long-memory processes, using an explicit representation of the finite predictor coefficients in terms of the autoregressive and moving-average coefficients. Due to non-commutativity of the matrix coefficients in the multivariate case, the techniques used in the univariate case are not longer valid. In this paper, using also the results from their paper [Proc. Am. Math. Soc. 144, No. 4, 1779--1786 (2016; Zbl 1338.60107)], under certain conditions new representation theorems are obtained which make possible to extend Baxter's inequality and other univariate asymptotic results to the multivariate long-memory processes. To do this, a nice presentation of preliminary notions and results is given, and then a projection theorem in terms of the Fourier coefficients of its phase function in the spectral domain is obtained. Using this, explicit representation theorems are obtained. The obtained representation theorems are used then to study a $q$-variate fractional autoregressive integrated moving-average process and to derive the asymptotics of the finite predictor error covariances and the partial autocorrelation function, as well as that of the finite predictor coefficients. Also, a Baxter's inequality for multivariate fractional autoregressive integrated moving-average processes is obtained, which extends the corresponding univariate result.