On linear processes with dependent innovations

Stochastic Processes and their Applications

Volumn 115, Issue 6, 2005, Pages 939-958

  Wu, Wei Biao ^a   Min, Wanli ^a,b  

^a UNIVERSITY OF CHICAGO   (United States)

^b IBM T J WATSON RESEARCH CENTER   (United States)

Author keywords

Central limit theorem; Covariance; GARCH model; Invariance principle; Linear process; Nonlinear time series

Indexed keywords

APPROXIMATION THEORY; ASYMPTOTIC STABILITY; BROWNIAN MOVEMENT; MATHEMATICAL MODELS; MATRIX ALGEBRA; PROBABILITY; THEOREM PROVING; TIME SERIES ANALYSIS; VECTORS;

CENTRAL LIMIT THEOREM; COVARIANCE; GARCH MODEL; LINEAR PROCESS; NONLINEAR TIME SERIES;

RANDOM PROCESSES;

EID: 17844387350     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.spa.2005.01.001     Document Type: Article

References (52)

1. D.W.K. Andrews, Nonstrong mixing autoregressive processes, J. Appl. Probab. 21 (4) (1984) 930-934.
2. R.T. Baillie, C.F. Chung, M.A. Tieslau, Analysing inflation by the fractionally integrated ARFIMA-GARCH model, J. Appl. Econom. 11 (1996) 23-40.
3. P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968.
4. N.H. Bingham, C.M. Goldie, J.M. Teugels, Regular Variation, Cambridge University Press, Cambridge, UK, 1987.
5. T. Bollerslev, Generalized autoregressive conditional heteroscedasticity, J. Econom. 31 (1986) 307-327.
6. P. Bougerol, N. Picard, Stationarity of GARCH processes and of some nonnegative time series, J. Econom. 52 (1992) 115-127.
7. G.E.P. Box, G.M. Jenkins, Time Series Analysis, Forecasting and Control, Holden-Day, San Francisco, 1976.
8. R. Bradley, Basic properties of strong mixing conditions, in: E. Eberlein M. Taqqu (Eds.) Dependence in Probability and Statistics: A Survey of Recent Results, Birkhauser, Boston, MA, 1986, pp. 165-192.
9. P.J. Brockwell, R.A. Davis, Time Series: Theory and Methods, Springer, New York, 1991.
10. M. Carrasco, X. Chen, Mixing and moment properties of various GARCH and stochastic volatility models, Econom. Theory 18 (2002) 17-39.
11. M. Chen, H. An, A note on the stationarity and existence of moments of the GARCH models, Statist. Sinica 8 (1998) 505-510.
12. Y.S. Chow, H. Teicher, Probability theory, second ed., Springer, New York, 1988.
13. C.F. Chung, Sample means, sample autocovariances, and linear regression of stationary multivariate long memory processes, Econom. Theory 18 (2002) 51-78.
14. Yu.A. Davydov, The invariance principle for stationary process, Theory Probab. Appl. 15 (1970) 487-498.
15. P. Diaconis, D. Freedman, Iterated random functions, SIAM Rev. 41 (1999) 41-76.
16. P.H. Diananda, Some probability limit theorems with statistical applications, Proc. Cambridge Philos. Soc. (1953) 239-246.
17. W. Feller, An introduction to probability theory and its applications, vol. II, Wiley, New York, 1971.
18. L. Giraitis, D. Surgailis, A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittle's estimate, Probab. Theory Related Fields 86 (1990) 87-104.
19. V.V. Gorodetskii, Convergence to semi-stable Gaussian processes, Theory Probab. Appl. 22 (1977) 498-508.
20. P. Hall, C.C. Heyde, Martingale Limit Theorem and its Application, Academic Press, New York, 1980.
21. E.J. Hannan, Central limit theorems for time series regression, Z. Wahrsch. Verw. Gebiete 26 (1973) 157-170.
22. E.J. Hannan, The asymptotic distribution of serial covariances, Ann. Statist. 4 (1976) 396-399.
23. E.J. Hannan, The central limit theorem for time series regression, Stochast. Process. Appl. 9 (1979) 281-289.
24. E.J. Hannan, C.C. Heyde, On limit theorems for quadratic functions of discrete time series, Ann. Math. Statist. 43 (1972) 2058-2066.
25. M.A. Hauser, R.M. Kunst, Forecasting high-frequency financial data with the ARFIMA-ARCH model, J. Forecasting 20 (2001) 501-518.
26. S. He, A note on the asymptotic normality of sample autocorrelations for a linear stationary sequence, J. Multivariate Anal. 58 (1996) 182-188.
27. C. He, T. Teräsvirta, Fourth moment structure of the GARCH ( p , q ) process, Econom. Theory 15 (1999) 824-846.
28. J.R.M. Hosking, Fractional differencing, Biometrika 73 (1981) 217-221.
29. J.R.M. Hosking, Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series, J. Econom. 73 (1996) 261-284.
30. T. Hsing, W.B. Wu, On weighted U-statistics for stationary processes, Ann. Probab. 32 (2003) 1600-1631.
31. O. Kallenberg, Foundations of Modern Probability, second ed., Springer, New York, 2002.
32. D. Lien Y.K. Tse, Forecasting the Nikkei spot index with fractional cointegration, J. Forecasting 18 (1999) 259-273.
33. D. Lien, Y.K. Tse, Fractional cointegration and futures hedging, J. Futures Markets 19 (1999) 457-474.
34. S. Ling, On the stationarity and the existence of moments of conditional heteroskedastic ARMA models, Statist. Sinica 9 (4) (1999) 1119-1130.
35. S. Ling, M. McAleer, Necessary and sufficient moment conditions for the GARCH ( r , s ) and asymmetric power GARCH (r, s) models, Econom. Theory 18 (2002) 722-729.
36. B. Mandelbrot, W.J. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (1969) 422-437.
37. D.L. McLeish, Invariance principles for dependent variables, Z. Wahrsch. Verw. Gebiete 32 (1975) 165-178.
38. D.L. McLeish, On the invariance principle for non-stationary mixingales, Ann. Probab. 5 (1977) 616-621.
39. S.P. Meyn, R.L. Tweedie, Markov Chains and Stochastic Stability, Springer, Berlin, 1994.
40. D.F. Nicholls, B.G. Quinn, Random Coefficient Autoregressive Models an Introduction, Springer, New York, 1982.
41. M. Peligrad, Recent advances in the central limit theorem and its weak invariance principle for mixing sequences of random variables (A survey), in: E. Eberlein M. Taqqu (eds.) Dependence in Probability and Statistics: A Survey of Recent Results, Birkhauser, Boston, MA, 1986, pp. 193-223.
42. P.C.B. Phillips, V. Solo, Asymptotics for linear processes, Ann. Statist. 20 (1992) 971-1001.
43. P.J. Romano, L.A. Thombs, Inference for autocorrelations under weak assumption, J. Amer. Statist. Assoc. 91 (1996) 590-600.
44. H. Tong, Non-linear Time Series: A Dynamic System Approach, Oxford University Press, Oxford, 1990.
45. Q.Y. Wang, Y.X. Lin, C.M. Gulati, The invariance principle for linear processes with applications, Econom. Theory 18 (2002) 119-139.
46. M. Woodroofe, A central limit theorem for functions of a Markov chain with applications to shifts, Stochast. Process. Appl. 41 (1992) 33-44.
47. W.B. Wu, Empirical processes of long-memory sequences, Bernoulli 9 (2003) 809-831.
48. W.B. Wu, Additive functionals of infinite-variance moving averages, Statist. Sinica 13 (2003) 1259-1267.
49. W.B. Wu, J. Mielniczuk, Kernel density estimation for linear processes, Ann. Statist. 30 (2002) 1441-1459.
50. W.B. Wu, M. Woodroofe, A central limit theorem for iterated random functions, J. Appl. Probab. 37 (2000) 748-755.
51. W.B. Wu, M. Woodroofe, Martingale approximations for sums of stationary processes, Ann. Probab. 32 (2004) 1674-1690.
52. R. Yokoyama, On the central limit theorem and law of the iterated logarithm for stationary processes with applications to linear processes, Stochast. Process. Appl. 59 (1995) 343-351.