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The integrated periodogram for long-memory processes with finite or infinite variance. (English) Zbl 0885.62108

The authors consider the stationary linear processes in the form \[ X_t=\sum _{j=0}^\infty c_jZ_{t-j}, \qquad t\in \mathcal {Z}, \] with a noise sequence \((Z_t)_{t\in \mathcal {Z}}\) of i.i.d. random variables which may have finite or infinite variance. The model may exhibit long-range dependence. The integrated periodogram \(K_n(\lambda )\) can be interpreted as the relative error of the empirical spectral density compared with the true spectral density in the interval \([0,\lambda ]\). The authors derive functional limit theorems for the randomly centered sequence \[ \left (K_n(\lambda )-K_n(\pi ){{\lambda +\pi }\over {2\pi }} \right )_{\lambda \in [-\pi ,\pi ]} \] The results are applied to obtain corresponding Kolmogorov–Smirnov and Cramér–von Mises goodness-of-fit tests.
Reviewer: G.Dohnal (Praha)

MSC:

62M15 Inference from stochastic processes and spectral analysis
60F17 Functional limit theorems; invariance principles
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G10 Nonparametric hypothesis testing

Software:

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References:

[1] Anderson, T. W., Goodness of fit tests for spectral distributions, Ann. Statist., 21, 830-847 (1993) · Zbl 0779.62083
[2] Bartlett, M. S., Problemes de l’analyse spectrale des séries temporelles stationnaires, Publ. Inst. Statist. Univ. Paris III-3, 119-134 (1954) · Zbl 0058.35704
[3] Bartlett, M. S., An Introduction to Stochastic Processes with Special Reference to Methods and Applications (1955), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0068.11801
[4] Beran, J., Statistics for Long-Memory Processes (1994), Chapman & Hall: Chapman & Hall New York · Zbl 0869.60045
[5] Billingsley, P., Convergence of Probability Measures (1968), Wiley: Wiley New York · Zbl 0172.21201
[6] Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation (1987), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0617.26001
[7] Brockwell, P. J.; Davis, R. A., Time Series: Theory and Methods (1991), Springer: Springer New York · Zbl 0709.62080
[8] Dahlhaus, R., Asymptotic normality of spectral estimates, J. Multivariate Anal., 16, 412-431 (1985) · Zbl 0579.62082
[9] Dahlhaus, R., Empirical spectral processes and their applications to time series analysis, Stochastic Process Appl., 30, 69-83 (1988) · Zbl 0655.60033
[10] Davis, R. A.; Resnick, S. I., Limit theory for moving averages of random variables with regularly varying tail probabilities, Ann. Probab., 13, 179-195 (1985) · Zbl 0562.60026
[11] Davis, R. A.; Resnick, S. I., Limit theory for the sample covariance and correlation functions of moving averages, Ann. Statist., 14, 533-558 (1986) · Zbl 0605.62092
[12] Feller, W., An Introduction to Probability Theory and its Applications II (1971), Wiley: Wiley New York · Zbl 0219.60003
[13] Giraitis, L.; Leipus, R., A functional central limit theorem for non-parametric estimates of spectra and the change-point problem for spectral functions, Lith. Math. Trans. (Lit. Mat. Sb.), 30, 674-697 (1990) · Zbl 0761.62123
[14] Giraitis, L.; Leipus, R., Testing and estimating in the change-point problem of the spectral function, Lith. Math. Trans. (Lit. Mat. Sb.), 32, 20-38 (1992) · Zbl 0794.62066
[15] L. Giraitis, R. Leipus and D. Surgailis, The change-point problem for dependent observations, J. Statist. Plann. Inference [To appear].; L. Giraitis, R. Leipus and D. Surgailis, The change-point problem for dependent observations, J. Statist. Plann. Inference [To appear]. · Zbl 0856.62073
[16] Giraitis, L.; Surgailis, D., A central limit theorem for the empirical process of a long memory linear sequence, Probab. Theory Related Fields (1994), [To appear]
[17] Grenander, U.; Rosenblatt, M., Statistical Analysis of Stationary Time Series (1984), Chelsea Publishing Co: Chelsea Publishing Co New York · Zbl 0575.62080
[18] Hida, T., Brownian Motion (1980), Springer: Springer New York · Zbl 0432.60002
[19] L. Horváth and P. Kokoszka, The effect of long-range dependence on change-point estimators, J. Statist. Plann. Inference [To appear].; L. Horváth and P. Kokoszka, The effect of long-range dependence on change-point estimators, J. Statist. Plann. Inference [To appear]. · Zbl 0946.62078
[20] C. Klüppelberg and T. Mikosch, The integrated periodogram for stable processes, Ann. Statist. [To appear].; C. Klüppelberg and T. Mikosch, The integrated periodogram for stable processes, Ann. Statist. [To appear]. · Zbl 0898.62116
[21] C. Klüppelberg and T. Mikosch, Gaussian limit fields for the integrated periodogram, Ann. Appl. Probab. [To appear].; C. Klüppelberg and T. Mikosch, Gaussian limit fields for the integrated periodogram, Ann. Appl. Probab. [To appear]. · Zbl 0866.60030
[22] Kokoszka, P. S., Prediction of infinite variance fractional ARIMA, Probab. Math. Statist., 16, 65-83 (1996) · Zbl 0857.60032
[23] Kokoszka, P. S.; Taqqu, M. S., Fractional ARIMA with stable innovations, Stochastic Process Appl., 60, 19-47 (1995) · Zbl 0846.62066
[24] P.S. Kokoszka and M.S. Taqqu, Parameter estimation for infinite variance fractional ARIMA, Ann. Statist. [To appear].; P.S. Kokoszka and M.S. Taqqu, Parameter estimation for infinite variance fractional ARIMA, Ann. Statist. [To appear]. · Zbl 0896.62092
[25] Kokoszka, P. S.; Taqqu, M. S., Discrete time parametric models with long memory and infinite variance (1996), Preprint · Zbl 0990.62080
[26] McConnell, T. R.; Taqqu, M. S., Decoupling inequalities for multilinear forms in independent symmetric random variables, Ann. Probab., 14, 943-954 (1986) · Zbl 0602.60025
[27] Mikosch, T., Functional limit theorems for random quadratic forms, Stochastic Process Appl., 37, 81-98 (1991) · Zbl 0726.60030
[28] Mikosch, T.; Gadrich, T.; Klüppelberg, C.; Adler, R. J., Parameter estimation for ARMA models with infinite variance innovations, Ann. Statist., 23, 305-326 (1995) · Zbl 0822.62076
[29] Mikosch, T.; Norvaiša, R., Uniform convergence of the empirical spectral distribution function (1995), Preprint · Zbl 0913.60032
[30] Priestley, M. B., (Spectral Analysis and Time Series I (1981), Academic Press: Academic Press New York) · Zbl 0744.62130
[31] Rosinski, J., Remarks on Banach spaces of stable type, Probab. Math. Statist., 1, 67-71 (1980) · Zbl 0511.60005
[32] Rosinski, J.; Woyczyński, W. A., Multilinear forms in Pareto-like random variables and product random measures, Coll. Math., 51, 303-313 (1987) · Zbl 0644.60016
[33] Samorodnitsky, G.; Taqqu, M. S., Stable Non-Gaussian Random Processes, (Stochastic Models with Infinite Variance (1994), Chapman and Hall: Chapman and Hall London) · Zbl 0925.60027
[34] Shorack, G. R.; Wellner, J. A., Empirical Processes with Applications to Statistics (1986), Wiley: Wiley New York · Zbl 1170.62365
[35] Zygmund, A., (Trigonometric Series (1988), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0628.42001
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