The integrated periodogram for long-memory processes with finite or infinite variance (Q1382496)

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.