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Teor. Veroyatnost. i Primenen., 2004, Volume 49, Issue 4, Pages 653–671 (Mi tvp187)  

This article is cited in 10 scientific papers (total in 10 papers)

On the central limit theorem for Toeplitz quadratic forms of stationary sequences

A. A. Sahakian, M. S. Ginovyan

Institute of Mathematics, National Academy of Sciences of Armenia

Abstract: Let $X(t)$, $t = 0,\pm1,\ldots$, be a real-valued stationary Gaussian sequence with a spectral density function $f(\lambda)$. The paper considers the question of applicability of the central limit theorem (CLT) for a Toeplitz-type quadratic form $Q_n$ in variables $X(t)$, generated by an integrable even function $g(\lambda)$. Assuming that $f(\lambda)$ and $g(\lambda)$ are regularly varying at $\lambda=0$ of orders $\alpha$ and $\beta$, respectively, we prove the CLT for the standard normalized quadratic form $Q_n$ in a critical case $\alpha+\beta=\frac{1}{2}$.
We also show that the CLT is not valid under the single condition that the asymptotic variance of $Q_n$ is separated from zero and infinity.

Keywords: stationary Gaussian sequence, spectral density, Toeplitz-type quadratic forms, central limit theorem, asymptotic variance, slowly varying functions.

DOI: https://doi.org/10.4213/tvp187

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English version:
Theory of Probability and its Applications, 2005, 49:4, 612–628

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Received: 17.05.2004

Citation: A. A. Sahakian, M. S. Ginovyan, “On the central limit theorem for Toeplitz quadratic forms of stationary sequences”, Teor. Veroyatnost. i Primenen., 49:4 (2004), 653–671; Theory Probab. Appl., 49:4 (2005), 612–628

Citation in format AMSBIB
\by A.~A.~Sahakian, M.~S.~Ginovyan
\paper On the central limit theorem for Toeplitz quadratic forms
of stationary sequences
\jour Teor. Veroyatnost. i Primenen.
\yr 2004
\vol 49
\issue 4
\pages 653--671
\jour Theory Probab. Appl.
\yr 2005
\vol 49
\issue 4
\pages 612--628

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    This publication is cited in the following articles:
    1. Ginovyan M.S., Sahakyan A.A., “Limit theorems for Toeplitz quadratic functionals of continuous-time stationary processes”, Probab. Theory Related Fields DS, 138:3-4 (2007), 551–579  crossref  mathscinet  zmath  isi  scopus
    2. Ginovyan M.S., Sahakyan A.A., “Error estimates for approximations of traces of products of truncated Toeplitz operators”, J. Contemp. Math. Anal., 43:4 (2008), 195–205  crossref  mathscinet  zmath  isi
    3. Ginovyan M.S., Sahakyan A.A., “A note on approximations of traces of products of truncated Toeplitz matrices”, J. Contemp. Math. Anal., 44:4 (2009), 262–269  crossref  mathscinet  zmath  isi  scopus
    4. Lavancier F., Philippe A., “Some convergence results on quadratic forms for random fields and application to empirical covariances”, Probab Theory Related Fields, 149:3–4 (2011), 493–514  crossref  mathscinet  zmath  isi  scopus
    5. Theory Probab. Appl., 56:1 (2012), 57–71  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. Ginovyan M.S. Sahakyan A.A., “On the Trace Approximations of Products of Toeplitz Matrices”, Stat. Probab. Lett., 83:3 (2013), 753–760  crossref  mathscinet  zmath  isi  scopus
    7. Ginovyan M.S. Sahakyan A.A., “On the Trace Approximation Problem for Truncated Toeplitz Operators and Matrices”, J. Contemp. Math. Anal.-Armen. Aca., 49:1 (2014), 1–16  crossref  mathscinet  zmath  isi  scopus
    8. Bai Sh., Ginovyan M.S., Taqqu M.S., “Functional Limit Theorems For Toeplitz Quadratic Functionals of Continuous Time Gaussian Stationary Processes”, Stat. Probab. Lett., 104 (2015), 58–67  crossref  mathscinet  zmath  isi  scopus
    9. Bai Sh., Ginovyan M.S., Taqqu M.S., “Limit theorems for quadratic forms of Lévy-driven continuous-time linear processes”, Stoch. Process. Their Appl., 126:4 (2016), 1036–1065  crossref  mathscinet  zmath  isi  elib  scopus
    10. Ginovyan M.S., Sahakyan A.A., “Limit Theorems For Tapered Toeplitz Quadratic Functionals of Continuous-Time Gaussian Stationary Processes”, J. Contemp. Math. Anal.-Armen. Aca., 54:4 (2019), 222–239  crossref  mathscinet  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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