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A multiplicative ergodic theorem for lipschitz maps

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  • Elton, John H.

Abstract

If (Fn, n [greater-or-equal, slanted] 0) is a stationary (ergodic) sequence of Lipschitz maps of a locally compact Polish space X into itself having a.s. negative Lyapunov exponent function, the composition process Fn...F1x converges in distribution to a stationary (ergodic) process in X (independent of x). For every x, the empirical distribution of a trajectory converges with probability one, and for every [var epsilon]>0, almost every trajectory is eventually within [var epsilon] of the support. We use the fact that the Lyapunov exponent of a process "run backwards" is the same as forwards. A set invariance condition is given for the case when (Fn) is a Markov chain. The result has applications to computer graphics and stability in control theory.

Suggested Citation

  • Elton, John H., 1990. "A multiplicative ergodic theorem for lipschitz maps," Stochastic Processes and their Applications, Elsevier, vol. 34(1), pages 39-47, February.
    • Handle: RePEc:eee:spapps:v:34:y:1990:i:1:p:39-47
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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. Mendivil, F., 2015. "Time-dependent iteration of random functions," Chaos, Solitons & Fractals, Elsevier, vol. 75(C), pages 178-184.
    7. 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.
    8. Fuh, Cheng-Der, 2021. "Asymptotic behavior for Markovian iterated function systems," Stochastic Processes and their Applications, Elsevier, vol. 138(C), pages 186-211.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. 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.
    15. 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.
    16. 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.
    17. Steinsaltz, David & Tuljapurkar, Shripad & Horvitz, Carol, 2011. "Derivatives of the stochastic growth rate," Theoretical Population Biology, Elsevier, vol. 80(1), pages 1-15.

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