The Annals of Probability

Limit theorems for functionals of moving averages

Hwai-Chung Ho and Tailen Hsing

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Let $X_n =\sum_{i=1}^\infty a_i \varepsilon_{n-i}$, where the $\varepsilon_i$ are i.i.d. with mean 0 and finite second moment and the $a_i$ are either summable or regularly varying with index $\in (-1,-1/2)$ . The sequence ${X_n}$ has short memory in the former case and long memory in the latter. For a large class of functions $K$, a new approach is proposed to develop both central ($\sqrt{N}$ rate) and noncentral (non-$\sqrt{N}$ rate) limit theorems for $S_N \equiv \sum_{n=1}^N [K(X_n) - EK (X_n)]$. Specifically, we show that in the short-memory case the central limit theorem holds for $S_N$ and in the long-memory case, $S_N$ can be decomposed into two asymptotically uncorrelated parts that follow a central limit and a non-central limit theorem, respectively. Further we write the noncentral part as an expansion of uncorrelated components that follow noncentral limit theorems. Connections with the usual Hermite expansion in the Gaussian setting are also explored.

Article information

Ann. Probab. Volume 25, Number 4 (1997), 1636-1669.

First available in Project Euclid: 7 June 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G10: Stationary processes

Asymptotic expansion central limit theorem fractional ARIMA process long memory long-range dependence noncentral limit theorem nonlinear function


Ho, Hwai-Chung; Hsing, Tailen. Limit theorems for functionals of moving averages. Ann. Probab. 25 (1997), no. 4, 1636--1669. doi:10.1214/aop/1023481106.

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  • Andrews, D. W. K. (1984). Non-strong mixing autoregressive processes. J. Appl. Probab. 21 930- 934.
  • Arcones, M. A. (1994). Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann. Probab. 22 2242-2274.
  • Athreya, K. B. and Pantula, S. G. (1986). A note on strong mixing of ARMA processes. Statist. Probab. Lett. 4 187-190.
  • Avram, F. and Taqqu, M. S. (1987). Noncentral limit theorems and Appell polynomials. Ann. Probab. 15 767-775.
  • Babu, G. J. and Singh, K. (1978). On deviations between the empirical and quantile processes for mixing random variables. J. Multivariate Anal. 8 532-549.
  • Basawa, I. V. and Prakasa Rao, B. L. S. (1980). Statistical Inference for Stochastic Processes. Academic Press, New York.
  • Beran, J. (1992). Statistical methods for data with long-range dependence. Statist. Sci. 7 404- 420.
  • Beran, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, New York.
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Bradley, R. C. (1986). Basic properties of strong mixing conditions. In Dependence in Probability and Statistics (E. Eberlein and M. S. Taqqu, eds.) 162-192. Birkh¨auser, Boston.
  • Breuer, P. and Major, P. (1983). Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Anal. 13 425-441.
  • Chanda, K. C. (1974). Strong mixing properties of linear stochastic processes. J. Appl. Probab. 11 401-408.
  • Chanda, K. C. and Ruymgaart, F. H. (1990). General linear processes: a property of the empirical process applied to density and mode estimation. J. Time Ser. Anal. 11 185-199.
  • Cox, D. R. (1984). Long-range dependence: a review. In Statistics: An Appraisal. Proceedings of the 50th Anniversary Conference (H. A. David and H. T. David, eds.) 55-74. Iowa State Univ. Press.
  • Cs ¨org o, S. and Mielniczuk, J. (1995). Density estimation under long-range dependence. Ann. Statist. 23 990-999.
  • Davydov, Y. A. (1970). The invariance principal for stationary processes. Theory Probab. Appl. 15 487-498.
  • Deo, C. M. (1973). A note on empirical processes for strong mixing sequences. Ann. Probab. 1 870-875.
  • Dobrushin, R. L. and Major, P. (1979). Non-central limit theorems for nonlinear functionals of Gaussian fields.Wahrsch. Verw. Gebiete 50 27-52.
  • Gastwirth, J. L. and Rubin, H. (1975). The asymptotic theory of the empirical cdf for mixing stochastic processes. Ann. Statist. 3 809-824.
  • Giraitis, L. (1985). Central limit theorem for functionals of a linear process. Lithuanian Math. J. 25 25-35.
  • Giraitis, L. and Surgailis, D. (1985). CLT and other theorems for functionals of Gaussian processes,Wahrsch. Verw. Gebiete 70 191-212.
  • Gorodetskii, V. V. (1977). On the strong mixing property for linear processes. Theory Probab. Appl. 22 411-413.
  • Granger, C. W. and Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. J. Time Ser. Anal. 1 15-29.
  • Hesse, C. H. (1990). Rates of convergence for the empirical distribution function and the empirical characteristic function of a broad class of linear processes. J. Multivariate Anal. 35 186-202.
  • Heyde, C. C. (1995). On the robustness of limit theorems. Bulletin of the ISI 50th session in Beijing, Book 2, 549-555.
  • Hipel, K. W. and McLeod, A. I. (1994). Time Series Modeling of Water Resources and Environmental Systems. North-Holland, Amsterdam.
  • Ho, H.-C. and Hsing, T. (1996). On the asymptotic expansion of the empirical process of longmemory moving averages. Ann. Statist. 24 992-1024.
  • Ho, H.-C. and Sun, T. C. (1987). A central limit theorem for non-instantaneous filters of a stationary Gaussian process. J. Multivariate Anal. 22 144-155.
  • Hosking, J. R. M. (1981). Fractional differencing. Biometrika 68 165-176.
  • K ¨unsch, H. (1986). Statistical aspects of self-similar processes. In Proceedings of the First World Congress of the Bernoulli Society, Tashkent 1 67-74.
  • Lai, T. L. and Stout, W. (1980). Limit theorems for sums of dependent random variables.Wahrsch. Verw. Gebiete 51 1-14.
  • Lo, A. W. (1991). Long-term memory in stock market prices. Econometrica 59 1279-1313.
  • Mehra, K. L. and Rao, M. S. (1975). Weak convergence of generalized empirical processes relative to dq under strong mixing. Ann. Probab. 3 979-991.
  • Pham, T. D. and Tran, T. T. (1985). Some mixing properties of time series models. Stochastic Process. Appl. 19 297-303.
  • Robinson, P. M. (1994). Time series with strong dependence. In Advances in Econometrics: Sixth World Congress 47-96. Cambridge Univ. Press.
  • Rosenblatt, M. (1984). Stochastic processes with short-range and long-range dependence. In Statistics: An Appraisal. Proceedings of the 50th Anniversary Conference (H. A. David and H. T. David, eds.) 509-520. Iowa State Univ. Press.
  • Sen, P. K. (1971). A note on weak convergence of empirical processes for sequences of -mixing random variables. Ann. Math. Statist. 42 2131-2133.
  • Silverman, B. W. (1983). Convergence of a class of empirical distribution functions of dependent random variables. Ann. Probab. 11 745-755.
  • Sun, T. C. and Ho, H.-C. (1985). Limit theorems of non-linear functions for stationary Gaussian Processes, In Dependence in Probability and Statistics (E. Eberlein and M. S. Taqqu, eds.) 1-17. Birkh¨auser, Boston.
  • Surgailis, D. (1982). Zones of attraction of self-similar multiple integrals. Lithuanian Math. J. 22 327-340.
  • Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank.Wahrsch. Verw. Gebiete 50 53-83.
  • Taqqu, M. S. (1985). A bibliographic guide to self-similar processes and long-range dependence. In Dependence in Probability and Statistics (E. Eberlein and M. S. Taqqu, eds.) 137- 165. Birkh¨auser, Boston.
  • Withers, C. S. (1975). Convergence of empirical processes of mixing random variables on 0 1. Ann. Statist. 3 1101-1108.