Limit theorems for functionals of moving averages. (English) Zbl 0903.60018

Summary: Let \(X_n= \sum^\infty_{i= 1}a_i\varepsilon_{n- i}\), 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 \(S_N\equiv \sum^N_{n= 1}[K(X_n)- EK(X_n)]\). 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 noncentral 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.


60F05 Central limit and other weak theorems
60G10 Stationary stochastic processes


Full Text: DOI


[1] Andrews, D. W. K. (1984). Non-strong mixing autoregressive processes. J. Appl. Probab. 21 930- 934. JSTOR: · Zbl 0552.60049
[2] Arcones, M. A. (1994). Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann. Probab. 22 2242-2274. · Zbl 0839.60024
[3] Athreya, K. B. and Pantula, S. G. (1986). A note on strong mixing of ARMA processes. Statist. Probab. Lett. 4 187-190. · Zbl 0596.60039
[4] Avram, F. and Taqqu, M. S. (1987). Noncentral limit theorems and Appell polynomials. Ann. Probab. 15 767-775. · Zbl 0624.60049
[5] 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. · Zbl 0395.62039
[6] Basawa, I. V. and Prakasa Rao, B. L. S. (1980). Statistical Inference for Stochastic Processes. Academic Press, New York. · Zbl 0448.62070
[7] Beran, J. (1992). Statistical methods for data with long-range dependence. Statist. Sci. 7 404- 420.
[8] Beran, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, New York. · Zbl 0869.60045
[9] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201
[10] 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. Birkhäuser, Boston. · Zbl 0603.60034
[11] Breuer, P. and Major, P. (1983). Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Anal. 13 425-441. · Zbl 0518.60023
[12] Chanda, K. C. (1974). Strong mixing properties of linear stochastic processes. J. Appl. Probab. 11 401-408. JSTOR: · Zbl 0281.60033
[13] 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. · Zbl 0719.62049
[14] 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.
[15] Cs örg o, S. and Mielniczuk, J. (1995). Density estimation under long-range dependence. Ann. Statist. 23 990-999. · Zbl 0843.62037
[16] Davydov, Y. A. (1970). The invariance principal for stationary processes. Theory Probab. Appl. 15 487-498. · Zbl 0219.60030
[17] Deo, C. M. (1973). A note on empirical processes for strong mixing sequences. Ann. Probab. 1 870-875. · Zbl 0281.60034
[18] Dobrushin, R. L. and Major, P. (1979). Non-central limit theorems for nonlinear functionals of Gaussian fields.Wahrsch. Verw. Gebiete 50 27-52. · Zbl 0397.60034
[19] Gastwirth, J. L. and Rubin, H. (1975). The asymptotic theory of the empirical cdf for mixing stochastic processes. Ann. Statist. 3 809-824. · Zbl 0318.62016
[20] Giraitis, L. (1985). Central limit theorem for functionals of a linear process. Lithuanian Math. J. 25 25-35. · Zbl 0568.60020
[21] Giraitis, L. and Surgailis, D. (1985). CLT and other theorems for functionals of Gaussian processes,Wahrsch. Verw. Gebiete 70 191-212. · Zbl 0575.60024
[22] Gorodetskii, V. V. (1977). On the strong mixing property for linear processes. Theory Probab. Appl. 22 411-413. · Zbl 0377.60046
[23] 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. · Zbl 0503.62079
[24] 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. · Zbl 0714.62044
[25] Heyde, C. C. (1995). On the robustness of limit theorems. Bulletin of the ISI 50th session in Beijing, Book 2, 549-555. · Zbl 0882.60019
[26] Hipel, K. W. and McLeod, A. I. (1994). Time Series Modeling of Water Resources and Environmental Systems. North-Holland, Amsterdam.
[27] Ho, H.-C. and Hsing, T. (1996). On the asymptotic expansion of the empirical process of longmemory moving averages. Ann. Statist. 24 992-1024. · Zbl 0862.60026
[28] 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. · Zbl 0623.60037
[29] Hosking, J. R. M. (1981). Fractional differencing. Biometrika 68 165-176. JSTOR: · Zbl 0464.62088
[30] K ünsch, H. (1986). Statistical aspects of self-similar processes. In Proceedings of the First World Congress of the Bernoulli Society, Tashkent 1 67-74.
[31] Lai, T. L. and Stout, W. (1980). Limit theorems for sums of dependent random variables.Wahrsch. Verw. Gebiete 51 1-14. · Zbl 0419.60026
[32] Lo, A. W. (1991). Long-term memory in stock market prices. Econometrica 59 1279-1313. · Zbl 0781.90023
[33] 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. · Zbl 0351.60032
[34] Pham, T. D. and Tran, T. T. (1985). Some mixing properties of time series models. Stochastic Process. Appl. 19 297-303. · Zbl 0564.62068
[35] Robinson, P. M. (1994). Time series with strong dependence. In Advances in Econometrics: Sixth World Congress 47-96. Cambridge Univ. Press.
[36] 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.
[37] Sen, P. K. (1971). A note on weak convergence of empirical processes for sequences of -mixing random variables. Ann. Math. Statist. 42 2131-2133. · Zbl 0226.60008
[38] Silverman, B. W. (1983). Convergence of a class of empirical distribution functions of dependent random variables. Ann. Probab. 11 745-755. · Zbl 0514.60040
[39] 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. Birkhäuser, Boston.
[40] Surgailis, D. (1982). Zones of attraction of self-similar multiple integrals. Lithuanian Math. J. 22 327-340. · Zbl 0515.60057
[41] Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank.Wahrsch. Verw. Gebiete 50 53-83. · Zbl 0397.60028
[42] 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. Birkhäuser, Boston. · Zbl 0596.60054
[43] Withers, C. S. (1975). Convergence of empirical processes of mixing random variables on 0 1. Ann. Statist. 3 1101-1108. · Zbl 0317.60013
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.