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Recent results in the theory and applications of CARMA processes

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Abstract

Just as ARMA processes play a central role in the representation of stationary time series with discrete time parameter, \((Y_n)_{n\in \mathbb {Z}}\), CARMA processes play an analogous role in the representation of stationary time series with continuous time parameter, \((Y(t))_{t\in \mathbb {R}}\). Lévy-driven CARMA processes permit the modelling of heavy-tailed and asymmetric time series and incorporate both distributional and sample-path information. In this article we provide a review of the basic theory and applications, emphasizing developments which have occurred since the earlier review in Brockwell (2001a, In D. N. Shanbhag and C. R. Rao (Eds.), Handbook of Statistics 19; Stochastic Processes: Theory and Methods (pp. 249–276), Amsterdam: Elsevier).

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Acknowledgments

Much of the recent material reviewed in this article is joint work with Alexander Lindner, to whom I am indebted for many valuable discussions. I am also indebted to two referees for their careful reading of the manuscript and many helpful comments.

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Authors and Affiliations

  1. Statistics Department, Colorado State University, Fort Collins, CO , 80523-1877, USA

    P. J. Brockwell

  2. Statistics Department, Columbia University, 1255 Amsterdam Avenue, New York, NY , 10027, USA

    P. J. Brockwell

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  1. P. J. Brockwell
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Correspondence to P. J. Brockwell.

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Support from the National Science Foundation Grant DMS-1107031 is gratefully acknowledged.

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Brockwell, P.J. Recent results in the theory and applications of CARMA processes. Ann Inst Stat Math 66, 647–685 (2014). https://doi.org/10.1007/s10463-014-0468-7

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  • DOI: https://doi.org/10.1007/s10463-014-0468-7

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