A Limit Theorem for Some Linear Processes with Innovations in the Domain of Attraction of a Stable Law

Abstract

Let $X=\{X_n: n\in {\mathbb {N}}\}$ be a linear process in which the coefficients are of the form $a_i=i^{-1}\ell (i)$ with $\ell $ being a slowly varying function at the infinity and the innovations are independent and identically distributed random variables belonging to the domain of attraction of an $\alpha $-stable law with $\alpha \in (1, 2]$. We will establish the asymptotic behavior of the partial sum process

$\begin{aligned} \bigg \{\sum \limits _{n=1}^{[Nt]} X_n: t\ge 0\bigg \} \end{aligned}$

as N tends to infinity, where [t] is the integer part of the nonnegative number t.

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Keywords

Limit theorem / Linear process / Domain of attraction of stable law / Convergence of finite-dimensional distributions / Infinite variance / 60F05 / 60G10 / 60E07 / 60E10

Funding

National Natural Science Foundation of China(12371156)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature