Persistence of one-dimensional AR(1)-processes

G. Hinrichs, M. Kolb, V. Wachtel, Journal of Theoretical Probability 33 (2018) 65–102.

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Journal Article | Published | English
Author
Hinrichs, Günter; Kolb, MartinLibreCat; Wachtel, Vitali
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Abstract
For a class of one-dimensional autoregressive sequences (Xn), we consider the tail behaviour of the stopping time T0=min{n≥1:Xn≤0}. We discuss existing general analytical approaches to this and related problems and propose a new one, which is based on a renewal-type decomposition for the moment generating function of T0 and on the analytical Fredholm alternative. Using this method, we show that Px(T0=n)∼V(x)Rn0 for some 0<R0<1 and a positive R0-harmonic function V. Further, we prove that our conditions on the tail behaviour of the innovations are sharp in the sense that fatter tails produce non-exponential decay factors.
Publishing Year
Journal Title
Journal of Theoretical Probability
Volume
33
Page
65–102
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Hinrichs G, Kolb M, Wachtel V. Persistence of one-dimensional AR(1)-processes. Journal of Theoretical Probability. 2018;33:65–102. doi:https://link.springer.com/article/10.1007/s10959-018-0850-0
Hinrichs, G., Kolb, M., & Wachtel, V. (2018). Persistence of one-dimensional AR(1)-processes. Journal of Theoretical Probability, 33, 65–102. https://link.springer.com/article/10.1007/s10959-018-0850-0
@article{Hinrichs_Kolb_Wachtel_2018, title={Persistence of one-dimensional AR(1)-processes}, volume={33}, DOI={https://link.springer.com/article/10.1007/s10959-018-0850-0}, journal={Journal of Theoretical Probability}, publisher={Springer Science + Business Media}, author={Hinrichs, Günter and Kolb, Martin and Wachtel, Vitali}, year={2018}, pages={65–102} }
Hinrichs, Günter, Martin Kolb, and Vitali Wachtel. “Persistence of One-Dimensional AR(1)-Processes.” Journal of Theoretical Probability 33 (2018): 65–102. https://link.springer.com/article/10.1007/s10959-018-0850-0.
G. Hinrichs, M. Kolb, and V. Wachtel, “Persistence of one-dimensional AR(1)-processes,” Journal of Theoretical Probability, vol. 33, pp. 65–102, 2018, doi: https://link.springer.com/article/10.1007/s10959-018-0850-0.
Hinrichs, Günter, et al. “Persistence of One-Dimensional AR(1)-Processes.” Journal of Theoretical Probability, vol. 33, Springer Science + Business Media, 2018, pp. 65–102, doi:https://link.springer.com/article/10.1007/s10959-018-0850-0.

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