Functional Convergence of Linear Processes with Heavy-Tailed Innovations

Raluca Balan (1) , Adam Jakubowski (2) , Sana Louhichi (3)

1 Department of Mathematics and Statistics [Ottawa]
2 Faculty of Mathematics and Computer Science [Toruń]
3 IPS - Inférence Processus Stochastiques

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 α-stable Lévy Motion are given. The conditions lead to new, tractable sufficient conditions in the case α ≤ 1. In the functional setting we complement the existing results on M 1-convergence, obtained for linear processes with nonnegative coefficients by Avram and Taqqu (1992) and improved by Louhichi and Rio (2011), by proving that in the general setting partial sums of linear processes are convergent on the Skorokhod space equipped with the S topology, introduced by Jakubowski (1997).

Mots clés: limit theorems; functional convergence; stable processes; linear processes

Domaines: Probabilités [math.PR]; Statistiques [math.ST]

Raluca Balan, Adam Jakubowski, Sana Louhichi. Functional Convergence of Linear Processes with Heavy-Tailed Innovations. Journal of Theoretical Probability, 2016, 29 (2), pp.491-526. ⟨10.1007/s10959-014-0581-9⟩. ⟨hal-01892125⟩