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Biometrika 2006 93(1):221-227; doi:10.1093/biomet/93.1.221
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© 2006 Biometrika Trust

Miscellanea

A note on time-reversibility of multivariate linear processes

Kung-Sik Chan1, Lop-Hing Ho2 and Howell Tong3

1 Department of Statistics and Actuarial Science, University of Iowa, Iowa City, Iowa 52242, U.S.A. kchan{at}stat.uiowa.edu, 2 Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas 67260, U.S.A. lop-hing.ho{at}wichita.edu, 3 Department of Statistics, London School of Economics, London WC2A 2AE, U.K. h.tong{at}lse.ac.uk

We derive some readily verifiable necessary and sufficient conditions for a multivariate non-Gaussian linear process to be time-reversible, under two sets of conditions on the contemporaneous dependence structure of the innovations. One set of 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 asymmetric 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 time-reversible if and only if the coefficients are essentially symmetric about some origin.

Key Words: Cumulant; Distributional equivalence; Non-Gaussian distribution; T-distribution; Time series; Symmetry

Received March 2005. Revised August 2005.


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