×

On the asymptotic expansion of the empirical process of long-memory moving averages. (English) Zbl 0862.60026

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 fourth moment and the \(a_i\) are regularly varying with index \(-\beta\) where \(\beta\in(1/2,1)\) so that \(\{X_n\}\) has long-range dependence. This covers an important class of the fractional ARIMA process. For \(r\geq0\), let \[ Y_{N,r}= \sum^N_{n=1} \sum_{1\leq j_1<\cdots<j_r}\prod^r_{s=1} a_{j_s}\varepsilon_{n-j_s},\quad Y_{N,0}=N,\quad \sigma^2_{N,r}=\text{Var}(Y_{N,r}) \] and \(F^{(r)}=\) the \(r\)th derivative of the distribution function of \(X_n\). The \(Y_{N,r}\) are uncorrelated and are stochastically decreasing in \(r\). For any positive integer \(p<(2\beta-1)^{-1}\), it is shown under mild regularity conditions that, with probability 1, \[ \sum^N_{n=1} I(X_n\leq x)= \sum^p_{r=0} (-1)^rF^{(r)}(x)Y_{N,r}+o(N^{-\lambda}\sigma_{N,p}) \] uniformly for all \(x\in{\mathfrak R}\), \(\forall 0<\lambda<(\beta-1/2)\wedge(1/2-p(\beta-1/2))\). This generalizes a host of existing results and provides the vehicle for a number of statistical applications.

MSC:

60G10 Stationary stochastic processes
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] AVRAM, F. and TAQQU, M. S. 1987. Noncentral limit theorems and Appell poly nomials. Ann. Probab. 15 767 775. Z. · Zbl 0624.60049 · doi:10.1214/aop/1176992170
[2] BAHADUR, R. R. 1966. A note on quantiles in large samples. Ann. Math. Statist. 37 577 580. Z. · Zbl 0147.18805 · doi:10.1214/aoms/1177699450
[3] BERAN, J. 1991. M-estimator of location for data with slowly decaying serial correlations. J. Amer. Statist. Assoc. 86 704 708. Z. JSTOR: · Zbl 0738.62082 · doi:10.2307/2290401
[4] BERAN, J. 1992. Statistical methods for data with long-range dependence. Statist. Sci. 7 404 420. Z.
[5] BERAN, J. and GHOSH, S. 1991. Slowly decaying correlation, testing normality, nuisance parameters. J. Amer. Statist. Assoc. 86 785 791. Z. JSTOR: · Zbl 0735.62084 · doi:10.2307/2290413
[6] BICKEL, P. and ROSENBLATT, M. 1973. On the global measure of the deviation of density function estimates. Ann. Statist. 1 1071 1095. Z. · Zbl 0275.62033 · doi:10.1214/aos/1176342558
[7] BILLINGSLEY, P. 1968. Convergence of Probability Measures. Wiley, New York. Z. · Zbl 0172.21201
[8] BROCKWELL, P. J. and DAVIS, R. A. 1987. Time Series: Theory and Methods. Springer, New York. Z. · Zbl 0604.62083
[9] CHANDA, K. C. and RUy MGAART, 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. Z. · Zbl 0719.62049 · doi:10.1111/j.1467-9892.1990.tb00051.x
[10] DAVy DOV, Y. A. 1970. The invariance principle for stationary processes. Theory Probab. Appl. 15 487 498. Z. · Zbl 0219.60030 · doi:10.1137/1115050
[11] DEHLING, H. and TAQQU, M. S. 1989. The empirical processes of some long-range dependent sequences with an application to U-statistics. Ann. Statist. 17 1767 1783. Z. · Zbl 0696.60032 · doi:10.1214/aos/1176347394
[12] DOBRUSHIN, R. L. and MAJOR, P. 1979. Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 27 52. Z. · Zbl 0397.60034 · doi:10.1007/BF00535673
[13] FELLER, W. 1971. An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York. · Zbl 0219.60003
[14] GRANGER, C. W. and JOy EUX, R. 1980. An introduction to long-memory time series models and fractional differencing. J. Time Ser. Anal. 1 15 29. Z. · Zbl 0503.62079 · doi:10.1111/j.1467-9892.1980.tb00297.x
[15] HALL, P. and HART, J. D. 1990. Convergence rates in density estimation for data from infinite-order moving average processes. Probab. Theory Related Fields 87 253 274. Z. · Zbl 0695.60043 · doi:10.1007/BF01198432
[16] HESSE, C. H. 1990a. 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. Z. · Zbl 0714.62044 · doi:10.1016/0047-259X(90)90023-B
[17] HESSE, C. H. 1990b. A Bahadur-ty pe representation for empirical quantiles of a large class of stationary, possibly infinite-variance, linear processes. Ann. Statist. 18 1188 1202. Z. · Zbl 0712.62042 · doi:10.1214/aos/1176347746
[18] HOSKING, J. R. M. 1981. Fractional differencing. Biometrika 68 165 176. Z. JSTOR: · Zbl 0464.62088 · doi:10.1093/biomet/68.1.165
[19] HURST, H. E. 1951. Long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers 116 770 808. Z.
[20] KUNSCH, H. 1986. Statistical aspects of self-similar processes. Proceedings of the First World \" Congress of the Bernoulli Society 1 67 74. Z. · Zbl 0673.62073
[21] LAI, T. L. and STOUT, W. 1980. Limit theorems for sums of dependent random variables. Z. Wahrsch. Verw. Gebiete 51 1 14. Z. · Zbl 0419.60026 · doi:10.1007/BF00533812
[22] MAJOR, P. 1981. Multiple Wiener Ito Integrals: With Applications to Limit Theorems. Lecture Notes in Math. 849. Springer, New York. Z. · Zbl 0451.60002 · doi:10.1007/BFb009403
[23] MANDELBROT, B. B. and TAQQU, M. S. 1979. Robust R S analysis of long run serial correlation. Proceedings of the 42nd Session of the ISI 2 69 100. Z. · Zbl 0518.62036
[24] POLLARD, D. 1984. Convergence of Stochastic Processes. Springer, New York. Z. · Zbl 0544.60045
[25] ROBINSON, P. M. 1994. Time series with strong dependence. Sixth World Congress of the Econometric Society 47 96. Cambridge Univ. Press. Z.
[26] ROSENBLATT, M. 1961. Independence and dependence. Proc. Fourth Berkeley Sy mp. Math. Statist. Probab. 2 431 443. Univ. California Press, Berkeley. Z. · Zbl 0105.11802
[27] SEN, P. K. 1972. On the Bahadur representation of sample quantiles for sequences of -mixing random variables. J. Multivariate Anal. 2 77 95. Z. · Zbl 0226.60050 · doi:10.1016/0047-259X(72)90011-5
[28] SURGAILIS, D. 1983. Zones of attraction of self-similar multiple integrals. Lithuanian Math. J. 22 327 340. Z. · Zbl 0537.60043 · doi:10.1007/BF00966427
[29] TAQQU, M. S. 1975. Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287 302. Z. · Zbl 0303.60033 · doi:10.1007/BF00532868
[30] TAQQU, M. S. 1979. Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50 53 83. Z. · Zbl 0397.60028 · doi:10.1007/BF00535674
[31] TAQQU, M. S. 1985. A bibliographic guide to self-similar processes and long-range dependence. Z. In Dependence in Probability and Statistics E. Eberlein and M. S. Taqqu, eds. 137 165. Birkhauser, Boston. \" · Zbl 0596.60054
[32] TAIPEI, TAIWAN 115 COLLEGE STATION, TEXAS 77843-3143 REPUBLIC OF CHINA E-MAIL: thsing@stat.tamu.edu
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.