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Central and Non-central Limit Theorems in a Free Probability Setting

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Abstract

Long-range dependence in time series may yield non-central limit theorems. We show that there are analogous time series in free probability with limits represented by multiple Wigner integrals, where Hermite processes are replaced by non-commutative Tchebycheff processes. This includes the non-commutative fractional Brownian motion and the non-commutative Rosenblatt process.

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Notes

  1. Cumulants have the following property which linearizes independence:

    $$ \kappa_n(X+Y,\ldots,X+Y)=\kappa_n(X, \ldots,X)+\kappa_n(Y,\ldots ,Y),\quad n\geq1. $$
    (2.7)

    Relation (2.7) holds in classical probability if X and Y are independent random variables and it holds in free probability if X and Y are freely independent (see [11, Proposition 12.3]). Since the classical notion of independence is different from the notion of free independence, the cumulants κ n in classical probability are different from those in free probability.

  2. Since all the free cumulants of S 2 are equal to 1, S 2 has a free Poisson law with mean 1.

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Acknowledgements

Ivan Nourdin was partially supported by the ANR Grants ANR-09-BLAN-0114 and ANR-10-BLAN-0121 at Université de Lorraine. Murad S. Taqqu was partially supported by the NSF Grant DMS-1007616 at Boston University.

We would like to thank two anonymous referees for their careful reading of the manuscript and for their valuable suggestions and remarks. Also, I. Nourdin would like to warmly thank M. S. Taqqu for his hospitality during his stay at Boston University in October 2011, where part of this research was carried out.

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Authors and Affiliations

  1. Faculté des Sciences et Techniques, Institut de Mathématiques Élie Cartan, Université de Lorraine, Campus Aiguillettes, B.P. 70239, 54506, Vandoeuvre-lès-Nancy Cedex, France

    Ivan Nourdin

  2. Fondation des Sciences Mathématiques de Paris, IHP, 11 rue Pierre et Marie Curie, 75231, Paris Cedex 05, France

    Ivan Nourdin

  3. Department of Mathematics, Boston University, 111 Cummington Road, Boston, MA, USA

    Murad S. Taqqu

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  1. Ivan Nourdin
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  2. Murad S. Taqqu
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Correspondence to Ivan Nourdin.

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Nourdin, I., Taqqu, M.S. Central and Non-central Limit Theorems in a Free Probability Setting. J Theor Probab 27, 220–248 (2014). https://doi.org/10.1007/s10959-012-0443-2

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