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Truncated moments of perpetuities and a new central limit theorem for GARCH processes without Kesten’s regularity

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  • Jakubowski, Adam
  • Szewczak, Zbigniew S.

Abstract

We consider a class of perpetuities which admit direct characterization of asymptotics of the key truncated moment. The class contains perpetuities without polynomial decay of tail probabilities thus not satisfying Kesten’s theorem. We show how to apply this result in deriving a new weak law of large numbers for solutions to stochastic recurrence equations and a new central limit theorem for GARCH(1,1) processes in the critical case.

Suggested Citation

  • Jakubowski, Adam & Szewczak, Zbigniew S., 2021. "Truncated moments of perpetuities and a new central limit theorem for GARCH processes without Kesten’s regularity," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 151-171.
    • Handle: RePEc:eee:spapps:v:131:y:2021:i:c:p:151-171
    DOI: 10.1016/j.spa.2020.09.003
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    References listed on IDEAS

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    1. 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.
    2. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    3. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    4. Basrak, Bojan & Davis, Richard A. & Mikosch, Thomas, 2002. "Regular variation of GARCH processes," Stochastic Processes and their Applications, Elsevier, vol. 99(1), pages 95-115, May.
    5. Bougerol, Philippe & Picard, Nico, 1992. "Stationarity of Garch processes and of some nonnegative time series," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 115-127.
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