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Persistence of One-Dimensional AR(1)-Sequences

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Abstract

For a class of one-dimensional autoregressive sequences \((X_n)\), we consider the tail behaviour of the stopping time \(T_0=\min \lbrace n\ge 1: X_n\le 0 \rbrace \). 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 \(T_0\) and on the analytical Fredholm alternative. Using this method, we show that \(\mathbb {P}_x(T_0=n)\sim V(x)R_0^n\) for some \(0<R_0<1\) and a positive \(R_0\)-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.

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Acknowledgements

The authors would like to thank a referee for her or his useful report and for suggestion to add the final section. This project was initiated during the sabbatical stay of VW at the Technion, Israel. He thanks the Humboldt foundation and the Technion for their financial support. MK would like to thank his student P. Trykacz for critical reading of the final draft.

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Correspondence to Vitali Wachtel.

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Hinrichs, G., Kolb, M. & Wachtel, V. Persistence of One-Dimensional AR(1)-Sequences. J Theor Probab 33, 65–102 (2020). https://doi.org/10.1007/s10959-018-0850-0

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Keywords

  • Persistence
  • Quasistationarity
  • Autoregressive sequence

Mathematics Subject Classification (2010)

  • Primary 60J05
  • Secondary 60G40