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

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Weak convergence of the function-indexed integrated periodogram for infinite variance processes. / Can, Umut ; Mikosch, Thomas Valentin; Samorodnitsky, Gennady .

In: Bernoulli, Vol. 16, No. 4, 2010, p. 995-1015.

Research output: Contribution to journal › Journal article › Research › peer-review

Harvard

Can, U, Mikosch, TV & Samorodnitsky, G 2010, 'Weak convergence of the function-indexed integrated periodogram for infinite variance processes', Bernoulli, vol. 16, no. 4, pp. 995-1015. https://doi.org/10.3150/10-BEJ253

APA

Can, U., Mikosch, T. V., & Samorodnitsky, G. (2010). Weak convergence of the function-indexed integrated periodogram for infinite variance processes. Bernoulli, 16(4), 995-1015. https://doi.org/10.3150/10-BEJ253

Vancouver

Can U, Mikosch TV, Samorodnitsky G. Weak convergence of the function-indexed integrated periodogram for infinite variance processes. Bernoulli. 2010;16(4):995-1015. https://doi.org/10.3150/10-BEJ253

Author

Can, Umut ; Mikosch, Thomas Valentin ; Samorodnitsky, Gennady . / Weak convergence of the function-indexed integrated periodogram for infinite variance processes. In: Bernoulli. 2010 ; Vol. 16, No. 4. pp. 995-1015.

Bibtex

@article{1d49fc0bc20a424594b4b3f9da68a4f1,
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",
year = "2010",
doi = "10.3150/10-BEJ253",
language = "English",
volume = "16",
pages = "995--1015",
journal = "Bernoulli",
issn = "1350-7265",
publisher = "International Statistical Institute",
number = "4",

}

RIS

TY - JOUR

T1 - Weak convergence of the function-indexed integrated periodogram for infinite variance processes

AU - Can, Umut

AU - Mikosch, Thomas Valentin

AU - Samorodnitsky, Gennady

PY - 2010

Y1 - 2010

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.

U2 - 10.3150/10-BEJ253

DO - 10.3150/10-BEJ253

M3 - Journal article

VL - 16

SP - 995

EP - 1015

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

IS - 4

ER -

ID: 33967663