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Limit theorems for Toeplitz quadratic functionals of continuous-time stationary processes
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  • Published: 17 November 2006

Limit theorems for Toeplitz quadratic functionals of continuous-time stationary processes

  • M. S. Ginovyan1 &
  • A. A. Sahakyan2 

Probability Theory and Related Fields volume 138, pages 551–579 (2007)Cite this article

Abstract

Let X(t), \(t\in\mathbb{R}\), be a centered real-valued stationary Gaussian process with spectral density f(λ). The paper considers a question concerning asymptotic distribution of Toeplitz type quadratic functional Q T of the process X(t), generated by an integrable even function g(λ). Sufficient conditions in terms of f(λ) and g(λ) ensuring central limit theorems for standard normalized quadratic functionals Q T are obtained, extending the results of Fox and Taqqu (Prob. Theory Relat. Fields 74: 213–240, 1987), Avram (Prob. Theory Relat. Fields 79:37–45, 1988), Giraitis and Surgailis (Prob. Theory Relat. Fields 86: 87–104, 1990), Ginovian and Sahakian (Theory Prob. Appl. 49:612–628, 2004) for discrete time processes.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, MCS 222, Boston University, 111 Cummington Street, Boston, MA, 02215, USA

    M. S. Ginovyan

  2. Institute of Mathematics of Armenian National Academy of Sciences, Marshal Bagramian Ave 24-B, 375019, Yerevan, Armenia

    A. A. Sahakyan

Authors
  1. M. S. Ginovyan
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  2. A. A. Sahakyan
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Correspondence to M. S. Ginovyan.

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Ginovyan, M.S., Sahakyan, A.A. Limit theorems for Toeplitz quadratic functionals of continuous-time stationary processes. Probab. Theory Relat. Fields 138, 551–579 (2007). https://doi.org/10.1007/s00440-006-0037-y

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  • Received: 26 May 2006

  • Revised: 21 September 2006

  • Published: 17 November 2006

  • Issue Date: July 2007

  • DOI: https://doi.org/10.1007/s00440-006-0037-y

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Keywords

  • Stationary Gaussian process
  • Spectral density
  • Toeplitz type quadratic functional
  • Central limit theorems

Mathematics Subject Classification (2000)

  • Primary 60G10
  • Primary 60F05
  • Secondary 60G15
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