A Note on Time-reversibility of Multivariate Linear Processes

We solve an important open problem by deriving some readily verifiable necessary and sufficient conditions for a multivariate non-Gaussian linear process to be time-reversible, under two sets of regularity conditions on the contemporaneous dependence structure of the innovations. One set of regularity conditions concerns the case of independent-component innovations, in which case a multivariate non-Gaussian linear process is time-reversible if and only if the coefficients consist of essentially symmetric columns with column-specific origins of symmetry or symmetric pairs of columns with pair-specific origins of symmetry. On the other hand, for dependent-component innovations plus other regularity conditions, a multivariate non-Gaussian linear process is timereversible if and only if the coefficients are essentially symmetric about some origin.