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Examples of such time series include stochastic processes in<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466622000172_inline1.png\"\/><jats:tex-math>$C[0,1]$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>equipped with the supremum distance and those in a finite dimensional vector space equipped with a non-Euclidean distance. We then develop versions of the Granger\u2013Johansen representation theorems for I(1) and I(2) autoregressive (AR) processes taking values in such a space. To achieve this goal, we first note that an AR(<jats:italic>p<\/jats:italic>) law of motion can be characterized by a linear operator pencil (an operator-valued map with certain properties) via the companion form representation, and then study the spectral properties of a linear operator pencil to obtain a necessary and sufficient condition for a given AR(<jats:italic>p<\/jats:italic>) law of motion to admit I(1) or I(2) solutions. These operator-theoretic results form a fundamental basis for our representation theorems. Furthermore, it is shown that our operator-theoretic approach is in fact a closely related extension of the conventional approach taken in a Euclidean space setting. Our theoretical results may be especially relevant in a recently growing literature on functional time series analysis in Banach spaces.<\/jats:p>","DOI":"10.1017\/s0266466622000172","type":"journal-article","created":{"date-parts":[[2022,4,20]],"date-time":"2022-04-20T08:41:43Z","timestamp":1650444103000},"page":"737-788","source":"Crossref","is-referenced-by-count":11,"title":["COINTEGRATION AND REPRESENTATION OF COINTEGRATED AUTOREGRESSIVE PROCESSES IN BANACH SPACES"],"prefix":"10.1017","volume":"39","author":[{"given":"Won-Ki","family":"Seo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2022,4,20]]},"reference":[{"key":"S0266466622000172_r42","unstructured":"Kheifets, I.L. & Phillips, P.C.B. (2022). Robust High-Dimensional IV Cointegration Estimation and Inference. 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