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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 $$ X_t=\sum _{j=0}^\infty c_jZ_{t-j}, \qquad t\in \Cal {Z},$$ 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.
Reviewer: G.Dohnal (Praha)

62M15Spectral analysis of processes
60F17Functional limit theorems; invariance principles
62M10Time series, auto-correlation, regression, etc. (statistics)
62G10Nonparametric hypothesis testing
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