On an autoregressive process driven by a sequence of Gaussian cylindrical random variables

Abstract

Let $\{Z_n\}_{n\in\mathbb{Z}}$ be a sequence of identically distributed, weakly independent and weakly Gaussian cylindrical random variables in a separable Banach space $U$. We consider the cylindrical difference equation, $X_n=AX_{n-1}+Z_n,~{n\in\mathbb{Z}}$, in $U$ and determine a cylindrical process $\{ Y_n\}_{n\in\mathbb{Z}}$ which solves the equation. The cylindrical distribution of $Y_n$ is shown to be weakly Gaussian and independent of $n$. It is also shown to be strongly Gaussian if the cylindrical distribution of $Z_1$ is strongly Gaussian. We determine the characteristic functional of $Y_n$ and give conditions under which $\{Y_n\}_{n\in\mathbb{Z}}$ is unique.