On solutions of a difference equation driven by a sequence of identically distributed and weakly independent cylindrical random variables

Abstract

We consider a sequence $\{Z_n\}_{n\in\Z}$ of weakly independent and identically distributed cylindrical random variables in a Banach space $U$ and a bounded linear operator $A$ on $U$ and show that under suitable conditions, for each $n\in\Z$, the series $\sum\limits_{k=0}^\infty Z_{n-k}((A^{\ast})^k(\cdot))$ converges in ${\mathfrak C}_2$, where ${\mathfrak C}_2$ is a Banach space of cylindrical random variables to be defined and if we define $Y_n:=\sum\limits_{k=0}^\infty Z_{n-k}((A^{\ast})^k(\cdot)) $, then the cylindrical process $\{Y_n\}_{n\in\Z}$ is the unique cylindrical weakly stationary solution of the cylindrical difference equation $X_n=AX_{n-1}+Z_n$. We show that without additional conditions on the operator $A$ and the cylindrical distribution of $Z_1$, the cylindrical distribution of $Y_n$ is $A$-decomposable. Further, we prove that under mild conditions on the cylindrical distribution of $Z_1$, $Y_n$ is induced by a $U$-valued random variable and finally we show that under certain conditions, the process $\{ X_n\}$ is a cylindrical Markov process.