Abstract
Most proofs in this chapter are more involved and are therefore omitted or simplified. For literature and detailed proofs see e.g. Berkes et al. (2003, 2004), Bollerslev (1986), Bougerol and Picard (1992a,b), Brandt (1986), Breiman (1968), Brockwell and Cline (1985), Caines (1988), Furstenberg and Kesten (1960), Giraitis et al. (2000), Hannan and Kanter (1977), Kazakevičius and Leipus (2002), Kingman (1973), Nelson (1990).
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References
Berkes, I., Horváth, L., & Kokoszka, P. (2003). GARCH processes: Structure and estimation. Bernoulli, 9, 201–227.
Berkes, I., Horváth, L., & Kokoszka, P. (2004). Probabilistic and statistical properties of GARCH processes. Fields Institute Communications, 44, 409–429.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, 307–327.
Bougerol, P., & Picard, N. (1992a). Strict stationarity of generalized autoregressive processes. The Annals of Probability, 20(4), 1714–1730.
Bougerol, P., & Picard, N. (1992b). Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics, 52, 115–127.
Brandt, A. (1986). The stochastic equation Y n+1 = A n Y n + B n with stationary coefficients. Advances in Applied Probability, 18, 211–220.
Breiman, L. (1968). Probability. Redding, MA: Addison-Wesley.
Brockwell, P. J., & Cline, D. B. H. (1985). Linear prediction of ARMA processes with infinite variance. Stochastic Processes and Applications, 19, 281–296.
Caines, P. E. (1988). Linear stochastic systems. New York: Wiley.
Furstenberg, H., & Kesten, H. (1960). Products of random matrices. Annals of Mathematical Statistics, 31, 457–469.
Giraitis, L., Kokoszka, P., & Leipus, R. (2000). Stationary ARCH models: Dependence structure and central limit theorem. Econometric Theory, 16, 3–22.
Hannan, E. J., & Kanter, M. (1977). Autoregressive processes with infinite variance. Journal of Applied Probability, 14, 411–415.
Kazakevičius, V., & Leipus, R. (2002). On stationarity of the ARCH(∞) model. Econometric Theory, 18(1), 1–16.
Kingman, J. F. C. (1973). Subadditive ergodic theory. The Annals of Probability, 1, 883–909.
Nelson, D. B. (1990). Stationarity and persistence in the GARCH(1,1) model. Econometric Theory, 6, 318–334.
Stout, H. F. (1974). Almost sure convergence. New York: Academic Press.
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Beran, J. (2017). Generalized Autoregressive Processes. In: Mathematical Foundations of Time Series Analysis. Springer, Cham. https://doi.org/10.1007/978-3-319-74380-6_7
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