## Bernoulli

• Bernoulli
• Volume 16, Number 4 (2010), 995-1015.

### Weak convergence of the function-indexed integrated periodogram for infinite variance processes

#### Abstract

In this paper, we study the weak convergence of the integrated periodogram indexed by classes of functions for linear processes with symmetric $α$-stable innovations. Under suitable summability conditions on the series of the Fourier coefficients of the index functions, we show that the weak limits constitute $α$-stable processes which have representations as infinite Fourier series with i.i.d. $α$-stable coefficients. The cases $α ∈ (0, 1)$ and $α ∈ [1, 2)$ are dealt with by rather different methods and under different assumptions on the classes of functions. For example, in contrast to the case $α ∈ (0, 1)$, entropy conditions are needed for $α ∈ [1, 2)$ to ensure the tightness of the sequence of integrated periodograms indexed by functions. The results of this paper are of additional interest since they provide limit results for infinite mean random quadratic forms with particular Toeplitz coefficient matrices.

#### Article information

Source
Bernoulli, Volume 16, Number 4 (2010), 995-1015.

Dates
First available in Project Euclid: 18 November 2010

https://projecteuclid.org/euclid.bj/1290092893

Digital Object Identifier
doi:10.3150/10-BEJ253

Mathematical Reviews number (MathSciNet)
MR2759166

Zentralblatt MATH identifier
1207.62173

#### Citation

Can, Sami Umut; Mikosch, Thomas; Samorodnitsky, Gennady. Weak convergence of the function-indexed integrated periodogram for infinite variance processes. Bernoulli 16 (2010), no. 4, 995--1015. doi:10.3150/10-BEJ253. https://projecteuclid.org/euclid.bj/1290092893

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