Convergence of averages of scaled functions of I(1) linear processes

Econometricians typically make use of functional central limit theorems to prove results for I (1) processes. For example, to establish the limit distributions of unit root tests such as the Phillips¯Perron and Dickey¯Fuller tests, the functional central limit theorem plays a crucial role. In this paper, it is pointed out that for linear processes, minimal conditions that ensure that only a central limit theorem holds are sufficient for establishing limit distributions of such tests. This eliminates the need to impose the stronger functional central limit theorem conditions and implies convergence of Dickey¯Fuller type unit root tests under minimal conditions.