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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.

DOI Code: 10.1285/i15900932v41n1p111

Keywords: Autoregressive process; Cylindrical process; Cylindrical measure; Cylindrical random variable; Stationary process

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