Central Limit Theorem for Linear Processes with Infinite Variance

TL;DR

This paper investigates the conditions under which linear processes with dependent, possibly infinite variance innovations satisfy the central limit theorem, extending classical results to more complex dependent structures and infinite variance cases.

Contribution

It establishes the equivalence between the CLT for linear processes and the domain of attraction of normal law for i.i.d. variables, and extends analysis to dependent variables with infinite variance.

Findings

CLT for linear processes is equivalent to variables being in the domain of attraction of normal law for i.i.d.

Results apply to dependent variables with infinite variance, relevant in economic models.

Addresses an open problem in the literature regarding dependent variables and infinite variance.

Abstract

This paper addresses the following classical question: giving a sequence of identically distributed random variables in the domain of attraction of a normal law, does the associated linear process satisfy the central limit theorem? We study the question for several classes of dependent random variables. For independent and identically distributed random variables we show that the central limit theorem for the linear process is equivalent to the fact that the variables are in the domain of attraction of a normal law, answering in this way an open problem in the literature. The study is also motivated by models arising in economic applications where often the innovations have infinite variance, coefficients are not absolutely summable, and the innovations are dependent.