Asymptotics for linear processes. (English) Zbl 0759.60021

Phillips, Peter C. B.; Solo, Victor

Ann. Stat. 20, No. 2, 971-1001 (1992).

A new method of deriving asymptotics for linear processes of the form \(X_ t=\sum c_ j\varepsilon_{t-j}\), where \(\{\varepsilon_ t\}\) is a sequence of time series innovations, is developed. The method is based on an algebraic decomposition of the linear filter into long-run and transitory components which is due to Beveridge and Nelson [J. Monetary Economics 7, 151-174 (1981)]. This method reduces a given limit problem for a linear process to the same problem for innovations \(\varepsilon_ t\) and to an estimation of some remainder. The authors present in a unified way a wide spectrum of limit results like laws of large numbers, laws of iterated logarithm, central limit theorems and invariance principles. The results accomodate both homogeneous and heterogeneous innovations as well as innovations with undefined means and variances. Besides results known from the literature some new theorems are also proved.

Reviewer: T.Inglot (Wrocław)

MSC:

60F05	Central limit and other weak theorems
60F17	Functional limit theorems; invariance principles
60F15	Strong limit theorems

Keywords:

stable processes; linear processes; algebraic decomposition of the linear filter; laws of large numbers; laws of iterated logarithm; central limit theorems; invariance principles