First order autoregressive periodically correlated model in Banach spaces: Existence and central limit theorem

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• 作者：A. Parvardeh^a ; ^{a.parvardeh@sci.ui.ac.ir} ; N. Mohammadi Jouzdani^b ; ^{neda.mohammadi@math.iut.ac.ir} ; S. Mahmoodi^b ; ^{mahmoodi@cc.iut.ac.ir} ; A.R. Soltani^c ; ^{soltani@kuc01.kuniv.edu.kw}
• 关键词：Periodically correlated process ; Linear time series ; Banach space ; Central limit theorem
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 May 2017
• 年：2017
• 卷：449
• 期：1
• 页码：756-768
• 全文大小：341 K
• 卷排序：449

文摘

We let BB be a separable Banach space, and let {Zn}{Zn} be a sequence of independent and identically distributed random elements in BB. Then we prove that for a given strongly periodic sequence of bounded linear operators {ρn}{ρn}, the order one autoregressive system equations Xn=ρnXn−1+Zn,n in set on integers, possesses a unique almost sure strictly periodically correlated solution; under E[log+⁡‖Z0‖]<∞E[log+⁡‖Z0‖]<∞, which appears to be necessary as well. We proceed on to derive the limiting distribution of ∑n=1NXn that appears to be a Gaussian distribution on BB. We also provide interesting examples and observations.