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Baxter's inequality and convergence of finite predictors of multivariate stochastic processess
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  • Published: March 1993

Baxter's inequality and convergence of finite predictors of multivariate stochastic processess

  • R. Cheng1 &
  • M. Pourahmadi2 

Probability Theory and Related Fields volume 95, pages 115–124 (1993)Cite this article

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Summary

We show that smoothness properties of a spectral density matrix and its optimal factor are closely related when the density satisfies theboundedness condition. This is crucial in proving multivariate generalizations of Baxter's inequality and obtaining rates of convergence of finite predictors. We rely on a technique of Lowdenslager and Rosenblum relating the optimal factor to the spectral density via Toeplitz operators.

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References

  1. Baxter, G.: A norm inequality for a “finite section” Wiener-Hopf equation. Ill. J. Math.7, 97–103 (1963)

    Google Scholar 

  2. Cheng, R.: Holder classes of vector-valued functions and convergence of the best predictor. J. Multivariate Anal.42, 110–129 (1992)

    Google Scholar 

  3. Devinatz, A.: Asymptotic estimates for the finite predictor. Math. Scand.15, 111–120 (1964)

    Google Scholar 

  4. Findley, D.: Convergence of finite multistep predictors from incorrect models and its role in model selection. Noti di Matematica (to appear 1993)

  5. Grenander, U., Rosenblatt, M.: An extension of a theorem of G. Szegö and its application to the study of stochastic processes. Trans. Am. Math. Soc.76, 112–126 (1954)

    Google Scholar 

  6. Grenander, U., Szegö, G.: Toeplitz forms and their applications. Berkeley: University of California Press 1958

    Google Scholar 

  7. Hannan, E.J., Diestler, M.: The statistical theory of linear systems. New York: Wiley 1988

    Google Scholar 

  8. Ibragimov, I.A.: On the asymptotic behavior of the prediction error. Theory Probab. Appl.9, 627–633 (1964)

    Google Scholar 

  9. Masani, P.R.: Recent trends in multivariate prediction theory. In: Krishnaiah, P.R. (ed.) Multivariate analysis, pp. 351–382, New York: Academic Press 1966

    Google Scholar 

  10. Miamee, A.G., Pourahmadi, M.: Degenerate multivariate stationary processes: Basicity, past and future, and autoregressive representation. Sankhyã49, 316–334 (1987)

    Google Scholar 

  11. Peller, V.V., Khrushchev, S.V.: Hankel operators, best approximations, and stationary Gaussian processes. Russ. Math. Surv.37, 61–144 (1982)

    Google Scholar 

  12. Pourahmadi, M.: On the convergence of finite linear predictors of stationary processes. J. Multivariate Anal.30, 167–180 (1989)

    Google Scholar 

  13. Rosenblum, M., Rovnyak, J.: Hardy classes and operator theory. New York: Oxford University Press 1985

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Louisville, 40292, Louisville, KY, USA

    R. Cheng

  2. Division of Statistics, Northern Illinois University, 60115-2854, Dekalb, IL, USA

    M. Pourahmadi

Authors
  1. R. Cheng
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  2. M. Pourahmadi
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Cite this article

Cheng, R., Pourahmadi, M. Baxter's inequality and convergence of finite predictors of multivariate stochastic processess. Probab. Th. Rel. Fields 95, 115–124 (1993). https://doi.org/10.1007/BF01197341

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  • Received: 12 December 1991

  • Revised: 27 July 1992

  • Issue Date: March 1993

  • DOI: https://doi.org/10.1007/BF01197341

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Mathematics Subject Classification (1991)

  • 60G25
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