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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 = j=0 c j Z t-j ,t𝒵,

with a noise sequence (Z t ) t𝒵 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 (λ) can be interpreted as the relative error of the empirical spectral density compared with the true spectral density in the interval [0,λ]. The authors derive functional limit theorems for the randomly centered sequence

K n (λ)-K n (π)λ+π 2π λ[-π,π]

The results are applied to obtain corresponding Kolmogorov–Smirnov and Cramér–von Mises goodness-of-fit tests.

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