Fredholm inversion around a singularity: Application to autoregressive time series in Banach space

Won-Ki Seo; Won-Ki Seo

doi:10.3934/era.2023252

Electronic Research Archive

2023, Volume 31, Issue 8: 4925-4950. doi: 10.3934/era.2023252

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Fredholm inversion around a singularity: Application to autoregressive time series in Banach space

Won-Ki Seo ^,

School of Economics, University of Sydney, Camperdown, NSW 2006, Australia

Received: 17 May 2023 Revised: 21 June 2023 Accepted: 30 June 2023 Published: 12 July 2023

This paper considers inverting a holomorphic Fredholm operator pencil. Specifically, we provide necessary and sufficient conditions for the inverse of a holomorphic Fredholm operator pencil to have a simple pole and a second order pole. Based on these results, a closed-form expression of the Laurent expansion of the inverse around an isolated singularity is obtained in each case. As an application, we also obtain a suitable extension of the Granger-Johansen representation theorem for random sequences taking values in a separable Banach space. Due to our closed-form expression of the inverse, we may fully characterize solutions to a given autoregressive law of motion except a term that depends on initial values.
- operator pencils,
- analytic Fredholm theorem,
- linear time-invariant dynamical systems,
- Granger-Johansen representation theorem
Citation: Won-Ki Seo. Fredholm inversion around a singularity: Application to autoregressive time series in Banach space[J]. Electronic Research Archive, 2023, 31(8): 4925-4950. doi: 10.3934/era.2023252

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Abstract

This paper considers inverting a holomorphic Fredholm operator pencil. Specifically, we provide necessary and sufficient conditions for the inverse of a holomorphic Fredholm operator pencil to have a simple pole and a second order pole. Based on these results, a closed-form expression of the Laurent expansion of the inverse around an isolated singularity is obtained in each case. As an application, we also obtain a suitable extension of the Granger-Johansen representation theorem for random sequences taking values in a separable Banach space. Due to our closed-form expression of the inverse, we may fully characterize solutions to a given autoregressive law of motion except a term that depends on initial values.

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