Functional central limit theorems for multivariate linear processes generated by dependent random vectors

Commun. Korean Math. Soc. 2006 Vol. 21, No. 4, 779-786 Published online December 1, 2006

Mi-Hwa Ko WonKwang University

Abstract : Let $\mathbb{X}_t$ be an $m$-dimensional linear process defined by $ \mathbb{X}_t = \sum_{j=0}^\infty A_j$ $\mathbb{Z}_{t-j},~ t=1,2,\ldots$, where $\{\mathbb{Z}_{t}\}$ is a sequence of $m$-dimensional random vectors with mean $\textbf{0}:m\times 1$ and positive definite covariance matrix $\Gamma:m\times m$ and $\{A_j\}$ is a sequence of coefficient matrices. In this paper we give sufficient conditions so that $\sum_{t=1}^{[ns]} \mathbb{X}_t$ (properly normalized) converges weakly to Wiener measure if the corresponding result for $\sum_{t=1}^{[n s]}\mathbb{Z}_t$ is true.

Keywords : functional central limit theorem, Linear process, moving average process, negatively associated, martingale difference