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Zbl 0885.62108
Kokoszka, P.; Mikosch, T.
The integrated periodogram for long-memory processes with finite or infinite variance.
(English)
[J] Stochastic Processes Appl. 66, No.1, 55-78 (1997). ISSN 0304-4149

The authors consider the stationary linear processes in the form $$X_t=\sum _{j=0}^\infty c_jZ_{t-j}, \qquad t\in \Cal {Z},$$ with a noise sequence $(Z_t)_{t\in \Cal {Z}}$ 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.
[G.Dohnal (Praha)]
MSC 2000:
*62M15 Spectral analysis of processes
60F17 Functional limit theorems
62M10 Time series, etc. (statistics)
62G10 Nonparametric hypothesis testing

Keywords: integrated periodogram; long-memory; heavy tails; functional limit theorems; goodness-of-fit tests; fractional ARIMA

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