Spectral analysis for amplitude-modulated time series. (English) Zbl 0787.62097

Toloi, Clélia M. C.; Morettin, Pedro A.

J. Time Ser. Anal. 14, No. 4, 409-432 (1993).

Let \(\{X(n)\}\) be a non-observed strictly stationary process with \(E| X(n)|^ k<\infty\), \(k>0\), a sequence independent of \(\{X(n)\}\), and \(Y(n)=a(n)X(n)\) the observed process. The problem of estimation of the spectral density \(f_ x(\lambda)\) is considered under assumptions \(A,B\) or \(C\), where:
A. \(\{a(n),n=0,\pm 1,\dots\}\) is a real deterministic sequence, asymptotically stationary, satisfying \(\sum_ n| a(n+1)- a(n)|<\infty\);
B. \(\{a(n),n=0,\pm 1,\dots\}\) and \(\{X(n),n=0,\pm 1,\dots\}\) are independent, strictly stationary processes with \(E| a(n)|^ k<\infty\);
C. \(\{a(n),\;n=0,\pm 1,\dots\}\) is a sequence of i.i.d. r.v. with nonzero mean \(\mu_ a\) and strictly positive variance \(\sigma^ 2_ a\), and \(\{X(n),n=0,\pm 1,\dots\}\) is independent of \(\{a(n)\}\).
The asymptotic normality of a finite Fourier transform \(d^{(N)}_ Y(\lambda)=\sum^{N=1}_{n=0}Y(n)e^{-i\lambda n}\) and consistent estimators of \(f_ X(\lambda)\), using weights of periodograms, are given.

Reviewer: N.Leonenko (Kiev)

MSC:

62M15	Inference from stochastic processes and spectral analysis
62F12	Asymptotic properties of parametric estimators
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)

Keywords:

amplitude-modulated time series; spectral estimation; strictly stationary process; spectral density; asymptotic normality; finite Fourier transform; consistent estimators; weights of periodograms

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