## On asymptotic distributions of weighted sums of periodograms.(English)Zbl 1280.62112

Summary: We establish the asymptotic normality of weighted sums of periodograms of a stationary linear process where the weights depend on the sample size. Such sums appear in numerous statistical applications and can be regarded as a discretized versions of quadratic forms involving integrals of weighted periodograms. Conditions for asymptotic normality of these weighted sums are simple, minimal, and resemble the Lindeberg-Feller condition for weighted sums of independent and identically distributed random variables. Our results are applicable to a large class of short, long or negative memory processes. The proof is based on sharp bounds derived for Bartlett type approximations of these sums by the corresponding sums of weighted periodograms of independent and identically distributed random variables.

### MSC:

 62M15 Inference from stochastic processes and spectral analysis 62E20 Asymptotic distribution theory in statistics
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### References:

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