Uniform convergence of the empirical spectral distribution function

Stochastic Processes and their Applications

Volumn 70, Issue 1, 1997, Pages 85-114

  Mikosch, T ^a   Norvaisa R ^b  

^a UNIVERSITY OF GRONINGEN   (Netherlands)

^b VILNIUS UNIVERSITY   (Lithuania)

Author keywords

Empirical spectral distribution function; Linear process; Periodogram; Spectral distribution function; Stationary sequence; Uniform central limit theorem; Uniform law of large numbers

Indexed keywords

EID: 0031256319     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-4149(97)00053-7     Document Type: Article

References (29)

1. Andersen, N.T., Dobrič, V., 1987. The central limit theorem for stochastic processes. Ann. Probab. 15, 164-177.
2. Anderson, T.W., 1993. Goodness of fit tests for spectral distributions. Ann. Statist. 21, 830-847.
3. Arcones, M.A., Giné, E., 1995. On the law of the iterated logarithm for canonical U-statistics and processes. Stochastic Processes Appl. 58, 217-246.
4. Bartlett, M.S., 1954. Problemes de l'analyse spectrale des séries temporelles stationnaires. Publ. Inst. Statist. Univ. Paris. III-3, 119-134.
5. Bartlett, M.S., 1978. An Introduction to Stochastic Processes with Special Reference to Methods and Applications, 3rd ed. Cambridge University Press, Cambridge, UK.
6. Billingsley, P., 1968. Convergence of Probability Measures. Wiley, New York.
7. Brillinger, D.R., 1981. Time Series. Data Analysis and Theory. Holden-Day, San Fracisco.
8. Brockwell, P.J., Davis, R.A., 1991. Time Series: Theory and Methods, 2nd ed. Springer, Berlin.
9. Chow, Y.S., Lai, T.L., 1973. Limiting behaviour of weighted sums of independent random variables. Ann. Probab. 1, 810-824.
10. Dahlhaus, R., 1988. Empirical spectral processes and their applications to time series analysis. Stochastic Processes Appl. 30, 69-83.
11. de la Pena, V.H., Montgomery-Smith, S.J., 1995. Decoupling inequalities for the tail probabilities of multivariate U-statistics. Ann. Probab. 23, 806-816.
12. Dudley, R.M., 1984. A Course on Empirical Processes. Lecture Notes in Math., Vol. 1097. Springer, Berlin.
13. Dzhaparidze, K., 1986. Parameter Estimation and Hypothesis Testing in Spectral Analysis of Stationary Time Series. Springer, Berlin.
14. Grenander, U., Rosenblatt, M., 1957. Statistical Analysis of Stationary Time Series. Wiley, New York.
15. Hosoya, Y., 1989. The bracketing condition for limit theorems on stationary linear processes. Ann. Statist. 17, 401-418.
16. Jacod, J., Shiryaev, A.N., 1987. Limit Theorems for Stochastic Processes. Springer, Berlin.
17. Klüppelberg, C, Mikosch, T., 1996. Gaussian limit fields for the integrated periodogram. Ann. Appl. Probab. 6, 969-991.
18. Kokoszka, P., Mikosch, T., 1997. The integrated periodogram for long-memory processes with finite or infinite variance. Stochastic Processes Appl. 66, 55-78.
19. Kwapień, S., Woyczyński, W.A., 1992. Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Basel.
20. Ledoux, M., Talagrand, M., 1991. Probability in Banach Spaces. Springer, Berlin.
21. Marcus, M.B., Zinn, J., 1984. The bounded law of the iterated logarithm for the weighted empirical distribution process in the non-i.i.d. case. Ann. Probab. 12, 335-360.
22. Mikosch, T., Norvaiša, R., 1987. Limit theorems for methods of summation of independent random variables I. Lit. Mat. Rink. (Lit. Mat. Sbor.; Lith. Math. J.) 27, 142-155.
23. Petrov, V.V., 1995. Limit Theorems of Probability Theory. Oxford University Press, Oxford.
24. Pisier, G., 1983. Some applications of the metric entropy condition to harmonic analysis. In: Banach spaces, Harmonic Analysis and Probability. University of Connecticut 1980-1981. Lecture Notes in Math., Vol. 995. Springer, Berlin, Heidelberg, pp. 123-154.
25. p (2 < p < ∞). Z. Wahrscheinlichkeitstheorie verw. Geb. 41, 289-309.
26. Pollard, D., 1984. Convergence of Stochastic Processes. Springer, Berlin.
27. Priestley, M.B., 1981. Spectral Analysis and Time Series. Academic Press, London.
28. van der Vaart, A.W., Wellner, J.A., 1996. Weak Convergence and Empirical Processes. Springer, New York.
29. Zygmund, A., 1988. Trigonometric Series. First paperback edition. Cambridge University Press, Cambridge, UK.