IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v16y2003i1d10.1023_a1022290807360.html
   My bibliography  Save this article

On the Harris Recurrence of Iterated Random Lipschitz Functions and Related Convergence Rate Results

Author

Listed:
  • Gerold Alsmeyer

Abstract

A result by Elton(6) states that an iterated function system $$M_n = F_n (M_{n - 1} ),{\text{ }}n \geqslant 1,$$ of i.i.d. random Lipschitz maps F 1,F 2,... on a locally compact, complete separable metric space $$(\mathbb{X},d)$$ converges weakly to its unique stationary distribution π if the pertinent Liapunov exponent is a.s. negative and $$\mathbb{E}\log ^ + d(F_1 (x_0 ),x_0 ) 0, where L 1 denotes the Lipschitz constant of F 1. The same and also polynomial rates have been recently obtained in Alsmeyer and Fuh(1) by different methods. In this article, necessary and sufficient conditions are given for the positive Harris recurrence of (M n ) n≥0 on some absorbing subset $$\mathbb{H}{\text{ of }}\mathbb{X}$$ . If $$\mathbb{H} = \mathbb{X}$$ and the support of π has nonempty interior, we further show that the same respective moment conditions ensuring the weak convergence rate results mentioned above now lead to polynomial, respectively geometric rate results for the convergence to π in total variation ∥⋅∥ or f-norm ∥⋅∥ f , f(x)=1+d(x,x 0) η for some η∈(0,p]. The results are applied to various examples that have been discussed in the literature, including the Beta walk, multivariate ARMA models and matrix recursions.

Suggested Citation

  • Gerold Alsmeyer, 2003. "On the Harris Recurrence of Iterated Random Lipschitz Functions and Related Convergence Rate Results," Journal of Theoretical Probability, Springer, vol. 16(1), pages 217-247, January.
    • Handle: RePEc:spr:jotpro:v:16:y:2003:i:1:d:10.1023_a:1022290807360
    DOI: 10.1023/A:1022290807360
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/A:1022290807360
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1023/A:1022290807360?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Elton, John H., 1990. "A multiplicative ergodic theorem for lipschitz maps," Stochastic Processes and their Applications, Elsevier, vol. 34(1), pages 39-47, February.
    2. Alsmeyer, Gerold & Fuh, Cheng-Der, 2001. "Limit theorems for iterated random functions by regenerative methods," Stochastic Processes and their Applications, Elsevier, vol. 96(1), pages 123-142, November.
    3. Niemi, S. & Nummelin, E., 1986. "On non-singular renewal kernels with an application to a semigroup of transition kernels," Stochastic Processes and their Applications, Elsevier, vol. 22(2), pages 177-202, July.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Szabolcs Blazsek & William M. Dos Santos & Andreco S. Edwards, 2024. "Score-Driven Interactions for “Disease X” Using COVID and Non-COVID Mortality," Econometrics, MDPI, vol. 12(3), pages 1-24, September.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Collamore, Jeffrey F. & Vidyashankar, Anand N., 2013. "Tail estimates for stochastic fixed point equations via nonlinear renewal theory," Stochastic Processes and their Applications, Elsevier, vol. 123(9), pages 3378-3429.
    2. Alsmeyer, Gerold, 2016. "On the stationary tail index of iterated random Lipschitz functions," Stochastic Processes and their Applications, Elsevier, vol. 126(1), pages 209-233.
    3. Fuh, Cheng-Der, 2021. "Asymptotic behavior for Markovian iterated function systems," Stochastic Processes and their Applications, Elsevier, vol. 138(C), pages 186-211.
    4. Dennis Kristensen, 2009. "On stationarity and ergodicity of the bilinear model with applications to GARCH models," Journal of Time Series Analysis, Wiley Blackwell, vol. 30(1), pages 125-144, January.
    5. Steinsaltz, David & Tuljapurkar, Shripad & Horvitz, Carol, 2011. "Derivatives of the stochastic growth rate," Theoretical Population Biology, Elsevier, vol. 80(1), pages 1-15.
    6. Olivier Wintenberger, 2013. "Continuous Invertibility and Stable QML Estimation of the EGARCH(1,1) Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(4), pages 846-867, December.
    7. Damek, Ewa & Kołodziejek, Bartosz, 2020. "Stochastic recursions: Between Kesten’s and Grincevičius–Grey’s assumptions," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1792-1819.
    8. Baye Matar Kandji, 2023. "On the growth rate of superadditive processes and the stability of functional GARCH models," Working Papers 2023-07, Center for Research in Economics and Statistics.
    9. Alsmeyer, Gerold & Hoefs, Volker, 2002. "Markov renewal theory for stationary (m+1)-block factors: convergence rate results," Stochastic Processes and their Applications, Elsevier, vol. 98(1), pages 77-112, March.
    10. Dmitrii S. Silvestrov & Örjan Stenflo, 1998. "Ergodic Theorems for Iterated Function Systems Controlled by Regenerative Sequences," Journal of Theoretical Probability, Springer, vol. 11(3), pages 589-608, July.
    11. Blazsek Szabolcs & Escribano Alvaro & Licht Adrian, 2021. "Identification of Seasonal Effects in Impulse Responses Using Score-Driven Multivariate Location Models," Journal of Econometric Methods, De Gruyter, vol. 10(1), pages 53-66, January.
    12. Mendivil, F., 2015. "Time-dependent iteration of random functions," Chaos, Solitons & Fractals, Elsevier, vol. 75(C), pages 178-184.
    13. Buraczewski, Dariusz & Damek, Ewa, 2017. "A simple proof of heavy tail estimates for affine type Lipschitz recursions," Stochastic Processes and their Applications, Elsevier, vol. 127(2), pages 657-668.
    14. Yuri Kabanov & Serguei Pergamenshchikov, 2020. "Ruin probabilities for a Lévy-driven generalised Ornstein–Uhlenbeck process," Finance and Stochastics, Springer, vol. 24(1), pages 39-69, January.
    15. Alsmeyer, Gerold & Fuh, Cheng-Der, 2001. "Limit theorems for iterated random functions by regenerative methods," Stochastic Processes and their Applications, Elsevier, vol. 96(1), pages 123-142, November.
    16. Blazsek, Szabolcs & Licht, Adrian, 2020. "Prediction accuracy of bivariate score-driven risk premium and volatility filters: an illustration for the Dow Jones," UC3M Working papers. Economics 31339, Universidad Carlos III de Madrid. Departamento de Economía.
    17. Kyprianou, Andreas E. & Palau, Sandra, 2018. "Extinction properties of multi-type continuous-state branching processes," Stochastic Processes and their Applications, Elsevier, vol. 128(10), pages 3466-3489.
    18. Roberts, Gareth O. & Rosenthal, Jeffrey S., 2002. "One-shot coupling for certain stochastic recursive sequences," Stochastic Processes and their Applications, Elsevier, vol. 99(2), pages 195-208, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:16:y:2003:i:1:d:10.1023_a:1022290807360. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.