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On the growth rate of superadditive processes and the stability of functional GARCH models

Author

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  • Baye Matar Kandji

    (CREST, ENSAE, Institut Polytechnique de Paris)

Abstract

We extend the result of Kesten (Proc. Am. Math. Soc., 49:205- 211, 1975) on the growth rate of random walks with stationary increments to superadditive processes. We show that superadditive processes which remain positive after a certain time diverge at least linearly to infinity. Our proof relies on new techniques based on concepts from ergodic theory. Different versions of this result are also given, generalizing Lemma 3.4 of Bougerol and Picard (Ann. Probab., 20:1714-1730, 1992) on the contraction property of products of random matrices. We use our results to provide necessary and sufficient conditions for the stability of a class of Stochastic Recurrent Equations (SRE) with positive coefficients in the space of continuous functions with compact support, including continuous functional GARCH models.

Suggested Citation

  • 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.
    • Handle: RePEc:crs:wpaper:2023-07
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    References listed on IDEAS

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    3. Cerovecki, Clément & Francq, Christian & Hörmann, Siegfried & Zakoïan, Jean-Michel, 2019. "Functional GARCH models: The quasi-likelihood approach and its applications," Journal of Econometrics, Elsevier, vol. 209(2), pages 353-375.
    4. Hörmann, Siegfried & Horváth, Lajos & Reeder, Ron, 2013. "A Functional Version Of The Arch Model," Econometric Theory, Cambridge University Press, vol. 29(2), pages 267-288, April.
    5. Elton, John H., 1990. "A multiplicative ergodic theorem for lipschitz maps," Stochastic Processes and their Applications, Elsevier, vol. 34(1), pages 39-47, February.
    6. He, Changli & Terasvirta, Timo, 1999. "Properties of moments of a family of GARCH processes," Journal of Econometrics, Elsevier, vol. 92(1), pages 173-192, September.
    7. Baye Matar Kandji, 2022. "Iterated Function Systems driven by non independent sequences: structure and inference," Working Papers 2022-03, Center for Research in Economics and Statistics.
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    More about this item

    Keywords

    Ergodic theorem Contraction property; functional Garch; Lyapunov exponent; Stochastic Recurrence Equation; Strict stationarity; Subadditive sequence.;
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