Symmetric and Reversed Multiple Stationary Autoregressive Series

Abstract

Let $\{X_t\}$ be a $p$-dimensional stationary autoregressive series. The main result is the determination of the autoregressive matrices of the series which is reversed in time with respect to $\{X_t\}$. The series which is reversed with respect to itself is called symmetric. The conditions for the symmetry of $\{X_t\}$ are given in the paper. The inverse of the covariance matrix is evaluated for the finite part of the symmetric autoregressive series.