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The Multifaceted Behavior of Integrated supOU Processes: The Infinite Variance Case

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Abstract

The so-called supOU processes, namely the superpositions of Ornstein–Uhlenbeck type processes, are stationary processes for which one can specify separately the marginal distribution and the temporal dependence structure. They can have finite or infinite variance. We study the limit behavior of integrated infinite variance supOU processes adequately normalized. Depending on the specific circumstances, the limit can be fractional Brownian motion but it can also be a process with infinite variance, a Lévy stable process with independent increments or a stable process with dependent increments. We show that it is even possible to have infinite variance integrated supOU processes converging to processes whose moments are all finite. A number of examples are provided.

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Acknowledgements

This work was supported by a grant from the Simons Foundation/569118MT at Boston University.

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Authors and Affiliations

  1. Department of Mathematics, University of Osijek, 31000, Osijek, Croatia

    Danijel Grahovac

  2. Cardiff School of Mathematics, Cardiff University, Cardiff, CF24 4AG, UK

    Nikolai N. Leonenko

  3. Department of Mathematics and Statistics, Boston University, Boston, MA, 02215, USA

    Murad S. Taqqu

Authors
  1. Danijel Grahovac
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  2. Nikolai N. Leonenko
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  3. Murad S. Taqqu
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Correspondence to Danijel Grahovac.

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Grahovac, D., Leonenko, N.N. & Taqqu, M.S. The Multifaceted Behavior of Integrated supOU Processes: The Infinite Variance Case. J Theor Probab 33, 1801–1831 (2020). https://doi.org/10.1007/s10959-019-00935-8

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  • DOI: https://doi.org/10.1007/s10959-019-00935-8

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