Journal of Statistical Physics

, Volume 177, Issue 4, pp 651–665 | Cite as

Persistence Exponents via Perturbation Theory: AR(1)-Processes

  • Frank Aurzada
  • Marvin KettnerEmail author


For AR(1)-processes \(X_n=\rho X_{n-1}+\xi _n\), \(n\in \mathbb {N}\), where \(\rho \in \mathbb {R}\) and \((\xi _i)_{i\in \mathbb {N}}\) is an i.i.d. sequence of random variables, we study the so-called persistence probabilities \(\mathbb {P}(X_0\ge 0,\ldots , X_N\ge 0)\) for \(N\rightarrow \infty \). For a wide class of Markov processes a recent result (Aurzada et al. in Persistence exponents in Markov chains, arXiv preprint arXiv:1703.06447, 2017) shows that these probabilities decrease exponentially fast and that the rate of decay can be identified as an eigenvalue of some integral operator. We discuss a perturbation technique to determine a series expansion of the eigenvalue in the parameter \(\rho \) for normally distributed AR(1)-processes.


Autoregressive process Eigenvalue problem Integral equation Markov chain Persistence Perturbation theory 



We would like to thank two anonymous referees for their comments that helped to improve the exposition of this paper. This work was supported by the Deutsche Forschungsgemeinschaft (Grant AU370/4).


  1. 1.
    Arendt, W., Batty, C.J., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics, vol. 96. Springer, Basel (2011)CrossRefGoogle Scholar
  2. 2.
    Aurzada, F., Baumgarten, C.: Survival probabilities of weighted random walks. ALEA, Lat. Am. J. Probab. Math. Stat. 8, 235–258 (2011)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Aurzada, F., Simon, T.: Persistence probabilities and exponents. In: Lévy matters. V, volume 2149 of Lecture Notes in Math., pp. 183–224. Springer, Cham (2015)Google Scholar
  4. 4.
    Aurzada, F., Mukherjee, S., Zeitouni, O.: Persistence exponents in Markov chains. arXiv preprint arXiv:1703.06447 (2017)
  5. 5.
    Baumgärtel, H.: Analytic Perturbation Theory for Matrices and Operators. Operator Theory: Advances and Applications, vol. 15. Birkhäuser, Basel (1985)zbMATHGoogle Scholar
  6. 6.
    Baumgarten, C.: Survival probabilities of autoregressive processes. ESAIM 18, 145–170 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bray, A., Majumdar, S.N., Schehr, G.: Persistence and first-passage properties in non-equilibrium systems. Adv. Phys. 62(3), 225–361 (2013)ADSCrossRefGoogle Scholar
  8. 8.
    Champagnat, N., Villemonais, D.: General criteria for the study of quasi-stationarity. arXiv preprint arXiv:1712.08092 (2017)
  9. 9.
    Collet, P., Martínez, S., San Martín, J.: Quasi-Stationary Distributions: Markov Chains, Diffusions and Dynamical Systems. Springer, NewYork (2012)zbMATHGoogle Scholar
  10. 10.
    Conway, J.B.: A Course in Functional Analysis. Graduate Texts in Mathematics, vol. 96. Springer, New York (2007)CrossRefGoogle Scholar
  11. 11.
    Dembo, A., Ding, J., Yan, J.: Persistence versus stability for auto-regressive processes. arXiv preprint arXiv:1906.00473 (2019)
  12. 12.
    Hinrichs, G., Kolb, M., Wachtel, V.: Persistence of one-dimensional AR(1)-sequences. arXiv preprint arXiv:1801.04485 (2018)
  13. 13.
    Janson, S.: Gaussian Hilbert Spaces, Cambridge Tracts in Mathematics, vol. 129. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  14. 14.
    Jentzsch, R.: Über Integralgleichungen mit positivem Kern. J. für die reine Angew. Math. 141, 235–244 (1912)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kato, T.: Perturbation Theory for Linear Operators. Die Grundlehren der mathematischen Wissenschaften, vol. 132. Springer, New York (1966)CrossRefGoogle Scholar
  16. 16.
    Letac, G.: Isotropy and sphericity: some characterisations of the normal distribution. Ann. Stat. 9(2), 408–417 (1981)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Majumdar, S.N., Dhar, D.: Persistence in a stationary time series. Phys. Rev. E 64, 046123 (2001)ADSCrossRefGoogle Scholar
  18. 18.
    Majumdar, S.N., Bray, A.J., Ehrhardt, G.C.: Persistence of a continuous stochastic process with discrete-time sampling. Phys. Rev. E 64, 015101(R) (2001)ADSCrossRefGoogle Scholar
  19. 19.
    Mehler, F.G.: Ueber die Entwicklung einer Function von beliebig vielen Variablen nach Laplaceschen Functionen höherer Ordnung. J. für die reine Angew Math. 66, 161–176 (1866)MathSciNetGoogle Scholar
  20. 20.
    Méléard, S., Villemonais, D.: Quasi-stationary distributions and population processes. Probab. Surv. 9, 340–410 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Metzler, R., Oshanin, G., Redner, S.: First-Passage Phenomena and Their Applications. World Scientific, Singapore (2014)CrossRefGoogle Scholar
  22. 22.
    Redner, S.: A Guide to First-Passage Processes. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  23. 23.
    Schaefer, H.H.: Banach Lattices and Positive Operators. Springer, Berlin (1974)CrossRefGoogle Scholar
  24. 24.
    Takesaki, M.: Theory of Operator Algebras I. Encyclopaedia of Mathematical Sciences, vol. 124. Springer, Berlin (2002)zbMATHGoogle Scholar
  25. 25.
    Tweedie, R.L.: Quasi-stationary distributions for Markov chains on a general state space. J. Appl. Probab. 11(4), 726–741 (1974)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Tweedie, R.L.: R-theory for Markov chains on a general state space I: solidarity properties and R-recurrent chains. Ann. Probab. 2(5), 840–864 (1974)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Technische Universität DarmstadtDarmstadtGermany

Personalised recommendations