Asymptotics for Linear Processes

TLDR

The technique extends Gordin's method, broadening its applicability to a wider range of linear processes. The study introduces a method for deriving asymptotics of linear processes using explicit algebraic decomposition of the linear filter. The method achieves this by algebraically decomposing the linear filter to obtain asymptotic results. The approach provides unified strong laws, central limit theorems, and invariance principles for linear processes, covering sample means and covariances and handling both homogeneous and heterogeneous innovations, including those with undefined means and variances.

Abstract

A method of deriving asymptotics for linear processes is introduced which uses an explicit algebraic decomposition of the linear filter. The technique is closely related to Gordin's method but has some advantages over it, especially in terms of its range of application. The method offers a simple unified approach to strong laws, central limit theory and invariance principles for linear processes. Sample means and sample covariances are covered. The results accommodate both homogeneous and heterogeneous innovations as well as innovations with undefined means and variances.