## Bernoulli

• Bernoulli
• Volume 19, Number 5B (2013), 2389-2413.

### On asymptotic distributions of weighted sums of periodograms

#### Abstract

We establish asymptotic normality of weighted sums of periodograms of a stationary linear process where 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 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 approximation of these sums by the corresponding sums of weighted periodograms of independent and identically distributed random variables.

#### Article information

Source
Bernoulli, Volume 19, Number 5B (2013), 2389-2413.

Dates
First available in Project Euclid: 3 December 2013

https://projecteuclid.org/euclid.bj/1386078607

Digital Object Identifier
doi:10.3150/12-BEJ456

Mathematical Reviews number (MathSciNet)
MR3160558

Zentralblatt MATH identifier
1280.62112

#### Citation

Giraitis, Liudas; Koul, Hira L. On asymptotic distributions of weighted sums of periodograms. Bernoulli 19 (2013), no. 5B, 2389--2413. doi:10.3150/12-BEJ456. https://projecteuclid.org/euclid.bj/1386078607

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