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Estimation of Mean and Covariance Operator for Banach Space Valued Autoregressive Processes with Dependent Innovations

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Abstract

In this paper we study autoregressive processes of order 1 with values in a separable Banach space it B. Such ARB(1)-processes \({(X_{n})}_{n \in \mathbb{Z}}\) are defined by the recursion equation

$$X_n - m = T(x_{n-1}-m) + \epsilon_n, n \in \mathbb{Z}$$

where T : B → B is a bounded linear operator and m ∈ B. We analyze the asymptotic properties of the sample mean and of the sample covariance operator in case that the innovation process \({(\epsilon_{n})}_{n \in \mathbb{Z}}\) is weakly dependent. This extends earlier results of Bosq (2000, 2002), who studied ARB(1)-processes with independent and orthogonal observations.

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References

  • V. Bentkus F. Götze A. Tikhomirov (1997) ArticleTitleBerry-Esseen bounds for statistics of weakly dependent samples Bernoulli 3 329–349 Occurrence Handle98m:60023

    MathSciNet  Google Scholar 

  • D. Bosq (2000) Linear Processes in Function Spaces. Theory and Applications, Lecture Notes in Statistics 149 Springer-Verlag New York

    Google Scholar 

  • D. Bosq (2002) ArticleTitleEstimation of mean and covariance operator of autoregressive processes in Banach spaces. Statist Infer. Stoch. Process. 5 287–306 Occurrence Handle1028.62070 Occurrence Handle2003m:60011

    MATH  MathSciNet  Google Scholar 

  • P. Doukhan (1994) Mixing, Properties and Examples, Lecture Notes in Statistics 85 Springer-Verlag New York

    Google Scholar 

  • I.A. Ibragimov (1962) ArticleTitleSome limit theorems for stationary processes Theor. Probab. Appl. 7 349–382 Occurrence Handle0119.14204

    MATH  Google Scholar 

  • M. Loève (1977) Probability Theory I Springer Verlag New York

    Google Scholar 

  • F. Merlevède M. Peligrad S. Utev (1997) ArticleTitleSharp conditions for the CLT of linear processes in a Hilbert space J. Theoret. Probab. 10 681–693 Occurrence Handle10.1023/A:1022653728014 Occurrence Handle99e:60022

    Article  MathSciNet  Google Scholar 

  • E. Mourier (1953) ArticleTitleElements Aleatoires dans un Espace de Banach Ann. de l' Institut Henri Poincare 13 159–244 Occurrence Handle16,268a

    MathSciNet  Google Scholar 

  • Q.M. Shao (1995) ArticleTitleMaximal inequalities for partial sums of .-mixing sequences Ann. Probab. 23 948–965 Occurrence Handle0831.60028 Occurrence Handle96d:60027

    MATH  MathSciNet  Google Scholar 

  • A. Tikhomirov (1991) ArticleTitleOn the accuracy of the normal approximation of the probability of sums of weakly dependent Hilbert-space-valued random variables hitting a ball Theor. Probab. Appl. 36 738–751 Occurrence Handle0776.60010 Occurrence Handle93e:60020

    MATH  MathSciNet  Google Scholar 

  • S. Utev (1991) ArticleTitleSums of random variables with .-mixing Siberian Adv. Math. 1 124–155 Occurrence Handle0733.60056 Occurrence Handle1128381

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, D-44780, Bochum, Germany

    Herold Dehling

  2. Department of Probability Theory, Institute of Mathematics, Uzbek Academy of Sciences, 29 F. Khodjaev str., Tashkent, 700143, Uzbekistan

    Olimjon SH. Sharipov

Authors
  1. Herold Dehling
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  2. Olimjon SH. Sharipov
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Correspondence to Herold Dehling.

Additional information

Research supported by DAAD (German Academic Exchange Service) grant A/01/26875.

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Dehling, H., Sharipov, O.S. Estimation of Mean and Covariance Operator for Banach Space Valued Autoregressive Processes with Dependent Innovations. Stat Infer Stoch Process 8, 137–149 (2005). https://doi.org/10.1007/s11203-003-0382-8

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  • DOI: https://doi.org/10.1007/s11203-003-0382-8

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