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

Umut  Can; Thomas Valentin Mikosch; Gennady  Samorodnitsky

doi:10.3150/10-BEJ253

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

Umut Can, Thomas Valentin Mikosch, Gennady Samorodnitsky

Institut for Matematiske Fag

3 Citationer (Scopus)

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.

Originalsprog	Engelsk
Tidsskrift	Bernoulli
Vol/bind	16
Udgave nummer	4
Sider (fra-til)	995-1015
Antal sider	21
ISSN	1350-7265
DOI	https://doi.org/10.3150/10-BEJ253
Status	Udgivet - nov. 2010

Adgang til dokumentet

10.3150/10-BEJ253

Citationsformater

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title = "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. ",

author = "Umut Can and Mikosch, {Thomas Valentin} and Gennady Samorodnitsky",

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AU - Can, Umut

AU - Mikosch, Thomas Valentin

AU - Samorodnitsky, Gennady

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N2 - 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.

AB - 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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