# zbMATH — the first resource for mathematics

Stationarity of GARCH processes and of some nonnegative time series. (English) Zbl 0746.62087
Summary: Two time series models are considered: GARCH processes and generalized multivariate autoregressive equations, $$X_{n+1}=A_{n+1}X_ n+B_{n+1}$$, with nonnegative i.i.d. coefficients. In each case, a necessary and sufficient condition ensuring the existence of a strictly stationary solution is given.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62P20 Applications of statistics to economics
Full Text:
##### References:
 [1] Bollerslev, T., Generalized autoregressive conditional heteroskedasticity, Journal of econometrics, 31, 307-327, (1986) · Zbl 0616.62119 [2] Bollerslev, T., A conditionally heteroskedastic time series model for security prices and rates of return data, Review of economics and statistics, 69, 542-547, (1987) [3] Bougerol, P., Tightness of products of random matrices and stability of linear stochastic systems, Annals of probability, 15, 40-74, (1987) · Zbl 0614.60008 [4] Bougerol, P.; Picard, N., Strict stationarity of generalized autoregressive processes, Annals of probability, (1990), forthcoming [5] Brandt, A., The stochastic equation Y_n+1=anyn+bn with stationary coefficients, Advances in applied probability, 18, 211-220, (1986) · Zbl 0588.60056 [6] Cohen, J.E.; Kesten, H.; Newman, C.M., Random matrices and their applications, () [7] Engle, R.F., Autoregressive conditional heteroskedasticity with estimates of the variance of the united kingdom inflation, Econometrica, 50, 987-1007, (1982) · Zbl 0491.62099 [8] Engle, R.F.; Bollerslev, T., Modelling the persistence of conditional variances, Econometric reviews, 5, 1-50, (1986) · Zbl 0619.62105 [9] Gaver, D.P.; Lewis, P.A.W., First-order autoregressive gamma sequences and point processes, Advances in applied probability, 12, 727-745, (1980) · Zbl 0453.60048 [10] Hutton, J.L., Non-negative time series models for dry river flow, Journal of applied probability, 27, 171-182, (1990) · Zbl 0699.62088 [11] Kesten, H.; Spitzer, F., Convergence in distribution for products of random matrices, Zeitschrift für wahrscheinlichkeitstheorie und verwandte gebiete, 67, 363-386, (1984) · Zbl 0535.60016 [12] Kingman, J.F.C., Subadditive ergodic theory, Annals of probability, 1, 883-909, (1973) · Zbl 0311.60018 [13] Lawrance, A.J.; Lewis, P.A.W., The exponential autoregressive moving average EARMA(p,q) process, Journal of the royal statistical society B, 42, 150-161, (1980) · Zbl 0433.62061 [14] Nelson, D.B., Stationarity and persistence in garch(1,1) model, Economic theory, (1988), forthcoming [15] Vervaat, W., On a stochastic difference equation and a representation of non-negative infinitely divisible random variables, Advances in applied probability, 11, 750-783, (1979) · Zbl 0417.60073
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.