Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Persistence of One-Dimensional AR(1)-Sequences

  • 46 Accesses

  • 1 Citations


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.

This is a preview of subscription content, log in to check access.


  1. 1.

    Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, Berlin (2006)

  2. 2.

    Aurzada, F., Mukherjee, S., Zeitouni, O.: Persistence exponents in Markov chains. ArXiv preprint: arXiv:1703.06447

  3. 3.

    Aurzada, F., Baumgarten, C.: Survival probabilities of weighted random walks. ALEA Lat. Am. J. Probab. Math. Stat. 8, 235–258 (2011)

  4. 4.

    Baumgarten, C.: Persistence of sums of independent random variables, iterated processes and fractional Brownian motion. Doctoral Thesis, Technical University Berlin (2013)

  5. 5.

    Champagnat, N., Villemonais, D.: Exponential convergence to quasi-stationary distributions and \(Q\)-process. Probab. Theory Relat. Fields 164, 243–283 (2016)

  6. 6.

    Champagnat, N., Villemonais, D.: General criteria for the study of quasi-stationarity. ArXiv preprint: arXiv:1712.08092

  7. 7.

    Christensen, S.: Phase-type distributions and optimal stopping for autoregressive processes. J. Appl. Prob. 49, 22–39 (2012)

  8. 8.

    Collet, P., Martinez, S., San Martin, J.: Quasi-Stationary Distributions: Markov Chains, Diffusions and Dynamical Systems. Springer, Berlin (2013)

  9. 9.

    Davies, E.B.: Linear Operators and Their Spectra. Cambridge University Press, Cambridge (2007)

  10. 10.

    Degla, G.: An overview of semi-continuity results on the spectral radius and positivity. J. Math. Anal. Appl. 338, 101–110 (2008)

  11. 11.

    Gosselin, F.: Aysmptotic behavior of absorbing markov chains conditional on nonabsorption for applications in conservation biology. Ann. Appl. Probab. 11, 261–284 (2001)

  12. 12.

    Hennion, H., Hervé, L.: Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Mathematics. Springer, Berlin (2000)

  13. 13.

    Lalley, S.P.: Return probabilities for random walk on a half-line. J. Theor. Probab. 8, 571–599 (1995)

  14. 14.

    Larralde, H.: A first passage time distribution for a discrete version of the Ornstein–Uhlenbeck process. J. Phys. A 37, 3759–3767 (2004)

  15. 15.

    Marek, I.: Spektrale Eigenschaften der \({\cal{K}}\)-positiven Operatoren und Einschließungssätze für den Spektralradius. Czechoslovak Math. J. 16, 493–517 (1966)

  16. 16.

    Meyer-Nieberg, P.: Banach Lattices. Springer, Universitätstext, Berlin (1991)

  17. 17.

    Novikov, A.: Some remarks on the distribution of the first passage times and the optimal stopping of AR(1)-sequences. Theory Probab. Appl. 53, 419–429 (2009)

  18. 18.

    Novikov, A., Kordzakhia, N.: Martingales and first passage times of AR(1) sequences. Stochastics 80, 197–210 (2008)

  19. 19.

    Revuz, D.: Markov Chains. North Holland Mathematical Library, Amsterdam (1984)

  20. 20.

    Sasser, D.W.: Quasi-positive operators. Pac. J. Math. 14, 1029–1037 (1964)

  21. 21.

    Schechter, M.: Principles of Functional Analysis. Academic Press, New York (1971)

  22. 22.

    Steinberg, S.: Meromorphic families of compact operators. Arch. Rational Mech. Anal. 31, 372–380 (1968)

  23. 23.

    Shumitzky, A., Wenska, T.: An operator residue theorem with applications to branching processes and renewal type integral equations. SIAM J. Math. Anal. 6, 229–235 (1975)

  24. 24.

    Tweedie, R.L.: R-theory for Markov chains on a general spate space I, soliditary properties and R-recurrent chains. Ann. Probab. 2, 840–864 (1974)

  25. 25.

    Tweedie, R.L.: R-theory for Markov chains on a general spate space II, r-subinvariant measures and r-transient chains. Ann. Probab. 2, 865–878 (1974)

  26. 26.

    Zerner, M.P.W.: Recurrence and transience of contractive autoregressive processes and related Markov chains. Electron. J. Probab. 23, 1–24 (2018)

Download references


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.

Author information

Correspondence to Vitali Wachtel.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hinrichs, G., Kolb, M. & Wachtel, V. Persistence of One-Dimensional AR(1)-Sequences. J Theor Probab 33, 65–102 (2020).

Download citation


  • Persistence
  • Quasistationarity
  • Autoregressive sequence

Mathematics Subject Classification (2010)

  • Primary 60J05
  • Secondary 60G40