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Persistence of One-Dimensional AR(1)-Sequences

Author

Listed:
  • Günter Hinrichs

    (Universität Augsburg)

  • Martin Kolb

    (Universität Paderborn)

  • Vitali Wachtel

    (Universität Augsburg)

Abstract

For a class of one-dimensional autoregressive sequences $$(X_n)$$(Xn), we consider the tail behaviour of the stopping time $$T_0=\min \lbrace n\ge 1: X_n\le 0 \rbrace $$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 $$T_0$$T0 and on the analytical Fredholm alternative. Using this method, we show that $$\mathbb {P}_x(T_0=n)\sim V(x)R_0^n$$Px(T0=n)∼V(x)R0n for some $$0

Suggested Citation

  • Günter Hinrichs & Martin Kolb & Vitali Wachtel, 2020. "Persistence of One-Dimensional AR(1)-Sequences," Journal of Theoretical Probability, Springer, vol. 33(1), pages 65-102, March.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:1:d:10.1007_s10959-018-0850-0
    DOI: 10.1007/s10959-018-0850-0
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    Cited by:

    1. Castro, Matheus M. & Lamb, Jeroen S.W. & Olicón-Méndez, Guillermo & Rasmussen, Martin, 2024. "Existence and uniqueness of quasi-stationary and quasi-ergodic measures for absorbing Markov chains: A Banach lattice approach," Stochastic Processes and their Applications, Elsevier, vol. 173(C).

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