Asymptotic normality of quadratic forms of martingale differences

Liudas Giraitis; Masanobu Taniguchi; Murad S. Taqqu

doi:10.1007/s11203-016-9143-3

Asymptotic normality of quadratic forms of martingale differences

Liudas Giraitis^*, Masanobu Taniguchi, Murad S. Taqqu

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

We establish the asymptotic normality of a quadratic form Q_n in martingale difference random variables η_t when the weight matrix A of the quadratic form has an asymptotically vanishing diagonal. Such a result has numerous potential applications in time series analysis. While for i.i.d. random variables η_t, asymptotic normality holds under condition | | A| | _s_p= o(| | A| |) , where | | A| | _s_p and ||A|| are the spectral and Euclidean norms of the matrix A, respectively, finding corresponding sufficient conditions in the case of martingale differences η_t has been an important open problem. We provide such sufficient conditions in this paper.

Original language	English
Pages (from-to)	315-327
Number of pages	13
Journal	Statistical Inference for Stochastic Processes
Volume	20
Issue number	3
DOIs	https://doi.org/10.1007/s11203-016-9143-3
Publication status	Published - 2017 Oct 1
Externally published	Yes

Keywords

Asymptotic normality
Martingale differences
Quadratic form

ASJC Scopus subject areas

Statistics and Probability

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10.1007/s11203-016-9143-3

Cite this

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title = "Asymptotic normality of quadratic forms of martingale differences",

abstract = "We establish the asymptotic normality of a quadratic form Qn in martingale difference random variables ηt when the weight matrix A of the quadratic form has an asymptotically vanishing diagonal. Such a result has numerous potential applications in time series analysis. While for i.i.d. random variables ηt, asymptotic normality holds under condition | | A| | sp= o(| | A| |) , where | | A| | sp and ||A|| are the spectral and Euclidean norms of the matrix A, respectively, finding corresponding sufficient conditions in the case of martingale differences ηt has been an important open problem. We provide such sufficient conditions in this paper.",

keywords = "Asymptotic normality, Martingale differences, Quadratic form",

author = "Liudas Giraitis and Masanobu Taniguchi and Taqqu, {Murad S.}",

note = "Funding Information: Liudas Giraitis and Murad S. Taqqu would like to thank Masanobu Taniguchi for his hospitality in Japan and support by the JSPS grant 15H02061. Murad S. Taqqu was partially supported by the NSF grant DMS-1309009 at Boston University. Publisher Copyright: {\textcopyright} 2016, The Author(s).",

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doi = "10.1007/s11203-016-9143-3",

language = "English",

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AU - Giraitis, Liudas

AU - Taniguchi, Masanobu

AU - Taqqu, Murad S.

N1 - Funding Information: Liudas Giraitis and Murad S. Taqqu would like to thank Masanobu Taniguchi for his hospitality in Japan and support by the JSPS grant 15H02061. Murad S. Taqqu was partially supported by the NSF grant DMS-1309009 at Boston University. Publisher Copyright: © 2016, The Author(s).

PY - 2017/10/1

Y1 - 2017/10/1

N2 - We establish the asymptotic normality of a quadratic form Qn in martingale difference random variables ηt when the weight matrix A of the quadratic form has an asymptotically vanishing diagonal. Such a result has numerous potential applications in time series analysis. While for i.i.d. random variables ηt, asymptotic normality holds under condition | | A| | sp= o(| | A| |) , where | | A| | sp and ||A|| are the spectral and Euclidean norms of the matrix A, respectively, finding corresponding sufficient conditions in the case of martingale differences ηt has been an important open problem. We provide such sufficient conditions in this paper.

AB - We establish the asymptotic normality of a quadratic form Qn in martingale difference random variables ηt when the weight matrix A of the quadratic form has an asymptotically vanishing diagonal. Such a result has numerous potential applications in time series analysis. While for i.i.d. random variables ηt, asymptotic normality holds under condition | | A| | sp= o(| | A| |) , where | | A| | sp and ||A|| are the spectral and Euclidean norms of the matrix A, respectively, finding corresponding sufficient conditions in the case of martingale differences ηt has been an important open problem. We provide such sufficient conditions in this paper.

KW - Asymptotic normality

KW - Martingale differences

KW - Quadratic form

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JO - Statistical Inference for Stochastic Processes

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Asymptotic normality of quadratic forms of martingale differences

Abstract

Keywords

ASJC Scopus subject areas

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