Functional Convergence of Linear Processes with Heavy-Tailed Innovations

Raluca Balan (), Adam Jakubowski () and Sana Louhichi ()
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Raluca Balan: University of Ottawa
Adam Jakubowski: Nicolaus Copernicus University
Sana Louhichi: Institut de mathématiques appliquées de Grenoble

Journal of Theoretical Probability, 2016, vol. 29, issue 2, 491-526

Abstract: Abstract We study convergence in law of partial sums of linear processes with heavy-tailed innovations. In the case of summable coefficients, necessary and sufficient conditions for the finite dimensional convergence to an $$\alpha $$ α -stable Lévy Motion are given. The conditions lead to new, tractable sufficient conditions in the case $$\alpha \le 1$$ α ≤ 1 . In the functional setting, we complement the existing results on $$M_1$$ M 1 -convergence, obtained for linear processes with nonnegative coefficients by Avram and Taqqu (Ann Probab 20:483–503, 1992) and improved by Louhichi and Rio (Electr J Probab 16(89), 2011), by proving that in the general setting partial sums of linear processes are convergent on the Skorokhod space equipped with the $$S$$ S topology, introduced by Jakubowski (Electr J Probab 2(4), 1997).

Keywords: Limit theorems; Functional convergence; Stable processes; Linear processes; 60F17; 60G52 (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10959-014-0581-9

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