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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)

##### MSC:
 62M15 Spectral analysis of processes 60F17 Functional limit theorems; invariance principles 62M10 Time series, auto-correlation, regression, etc. (statistics) 62G10 Nonparametric hypothesis testing
longmemo
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