Asymptotically homogeneous iterated random functions with applications to the HARCH process. (English) Zbl 1193.60092
Iterative time series on a finite dimensional space are considered of the form \(X_n=F(X_{n-1},\varepsilon_n)\), \(X_0=x\), where \(\varepsilon_i\) are i.i.d., \(F(\cdot,w)\) is an asymptotically homogeneous function, i.e. \(F(t_n x_n,w)/t_n\to G(x,w)\) as \(t_n\to\infty\), \(x_n\to x\), \(G\) is some function.
The authors establish conditions under which the Markov chain \(X_n\) is positive Harris recurrent. The key condition is \[ \lim_{m\to\infty}{1\over m}\sum_{i=1}^m E\ln\|G(Z_{i-1},\varepsilon_i)\|<0, \] where the auxiliary chain \(Z_i\) is defined by \(Z_i=G(Z_{i-1},\varepsilon_i)/\|G(Z_{i-1},\varepsilon_i)\|\).
This result is applied to derive conditions of stationarity of HARCH(1) process.
The authors establish conditions under which the Markov chain \(X_n\) is positive Harris recurrent. The key condition is \[ \lim_{m\to\infty}{1\over m}\sum_{i=1}^m E\ln\|G(Z_{i-1},\varepsilon_i)\|<0, \] where the auxiliary chain \(Z_i\) is defined by \(Z_i=G(Z_{i-1},\varepsilon_i)/\|G(Z_{i-1},\varepsilon_i)\|\).
This result is applied to derive conditions of stationarity of HARCH(1) process.
Reviewer: R. E. Maiboroda (Kyïv)
MSC:
| 60J05 | Discrete-time Markov processes on general state spaces |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 60G10 | Stationary stochastic processes |
Keywords:
Markov chain; stationary distribution; Foster-Lyapunov condition; HARCH process; Harris recurrenceReferences:
| [1] | P. Bougerol and N. Picard, Stationarity of GARCH processes and of some nonnegative time series, J. Econometrics, 52:115–127, 1992. · Zbl 0746.62087 · doi:10.1016/0304-4076(92)90067-2 |
| [2] | L. Breiman, The strong law of large numbers for a class of Markov chains, Ann. Math. Stat., 31:801–803, 1960. · Zbl 0104.11901 · doi:10.1214/aoms/1177705810 |
| [3] | P. Embrechts, G. Samorodnitsky, M.M. Dacorogna, and U.A. Muller, How heavy are the tails of a stationary HARCH(k) process? A study of the moments, in B.S. Rajput I. Karatzas and M.S. Taqqu (Eds.), Stochastic Processes and Related Topics: in Memory of Stamatis Cambanis 1943–1995, Birkhauser Verlag AG, 1998, pp. 69–102. · Zbl 0914.60065 |
| [4] | R.F. Engle, Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica, 50:987–1008, 1982. · Zbl 0491.62099 · doi:10.2307/1912773 |
| [5] | L. Giraitis, R. Leipus, and D. Surgailis, Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, in A. Kirman and G. Teyssiere (Eds.), Long Memory in Economics, New York, 2007, pp. 3–38. |
| [6] | S.P. Meyn and L. R Tweedie, Markov Chains and Stochastic Stability, Springer-Verlag, London, 1996. · Zbl 0925.60001 |
| [7] | U.A. Muller, M.M. Dacorogna, J.E. von Weizsacker, R.D. Dave, R.B. Olsen, and O.V. Pictet, Volatilities of different time resolutions – analyzing the dynamics of market components, J. Emp. Fin., 4:213–239, 1997. · doi:10.1016/S0927-5398(97)00007-8 |
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