# Functional Convergence of Linear Processes with Heavy-Tailed Innovations

Journal of Theoretical Probability, Oct 2014

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 $\alpha$-stable Lévy Motion are given. The conditions lead to new, tractable sufficient conditions in the case $\alpha \le 1$. In the functional setting, we complement the existing results on $M_1$-convergence, obtained for linear processes with nonnegative coefficients by Avram and Taqqu (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 $S$ topology, introduced by Jakubowski (Electr J Probab 2(4), 1997).

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Raluca Balan, Adam Jakubowski, Sana Louhichi. Functional Convergence of Linear Processes with Heavy-Tailed Innovations, Journal of Theoretical Probability, 2016, 491-526, DOI: 10.1007/s10959-014-0581-9