Time series: theory and methods
Time series: theory and methods
Journal of Multivariate Analysis
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A multivariate analogue of the fractionally integrated continuous time autoregressive moving average (FICARMA) process defined by Brockwell [Representations of continuous-time ARMA processes, J. Appl. Probab. 41 (A) (2004) 375-382] is introduced. We show that the multivariate FICARMA process has two kernel representations: as an integral over the fractionally integrated CARMA kernel with respect to a Levy process and as an integral over the original (not fractionally integrated) CARMA kernel with respect to the corresponding fractional Levy process (FLP). In order to obtain the latter representation we extend FLPs to the multivariate setting. In particular we give a spectral representation of FLPs and consequently, derive a spectral representation for FICARMA processes. Moreover, various probabilistic properties of the multivariate FICARMA process are discussed. As an example we consider multivariate fractionally integrated Ornstein-Uhlenbeck processes.