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November 2010 Weak convergence of the function-indexed integrated periodogram for infinite variance processes
Sami Umut Can, Thomas Mikosch, Gennady Samorodnitsky
Bernoulli 16(4): 995-1015 (November 2010). DOI: 10.3150/10-BEJ253


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.


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Sami Umut Can. Thomas Mikosch. Gennady Samorodnitsky. "Weak convergence of the function-indexed integrated periodogram for infinite variance processes." Bernoulli 16 (4) 995 - 1015, November 2010.


Published: November 2010
First available in Project Euclid: 18 November 2010

zbMATH: 1207.62173
MathSciNet: MR2759166
Digital Object Identifier: 10.3150/10-BEJ253

Rights: Copyright © 2010 Bernoulli Society for Mathematical Statistics and Probability


Vol.16 • No. 4 • November 2010
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