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On some extremal problems in Hp and the prediction ofL p-harmonizable stochastic processes
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  • Published: June 1994

On some extremal problems in Hp and the prediction ofL p-harmonizable stochastic processes

  • Balram S. Rajput1 &
  • Carl Sundberg1 

Probability Theory and Related Fields volume 99, pages 197–210 (1994)Cite this article

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Summary

We providesimple andsuccinct solutions to two dual extremal problems in the Hardy spacesH p, and to an aspect of the linear prediction problem for a certain class of discrete and continuous parameter “L p-harmonizable” stochastic processes, for all 1≦p<∞. Two of the results presented appear new. The methods of proof of the rest of the results provide alternatesimpler andshorter proofs for some earlier known theorems.

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References

  1. Cambanis, S., Soltani, A.R.: Prediction of stable processes: Spectral and moving average representations. Z. Wahrscheinlichkeitstheor. Verw. Geb.66, 593–612 (1984)

    Google Scholar 

  2. Cambanis, S., Hardin Jr, C.D., Weron, A.: Innovations and Wold decompositions of stable sequences. Probab. Theory Relat. Fields79, 1–27 (1988)

    Google Scholar 

  3. Cambanis, S., Miamee, A.G.: On prediction of harmonizable stable processes. Sankhya, Ser.A51, 269–294 (1989)

    Google Scholar 

  4. Duren, P.L.: Theory of Hp Spaces. New York: Academic Press 1970

    Google Scholar 

  5. Gihman, I.I., Skorohod, A.V.: The theory of stochastic processes I. Berlin Heidelberg New York: Springer 1974

    Google Scholar 

  6. Garnett, J.B.: Bounded analytic functions. New York: Academic Press 1981

    Google Scholar 

  7. Hosoya, Y.: Harmonizable stable processes. Z. Warscheinlichkeitstheor. Verw. Geb.60, 517–533 (1982)

    Google Scholar 

  8. Kallianpur, G.: Review of the book Stationary sequences and random fields. Rosenblatt, M. Bull. Am. Math. Soc.21, 133–139 (1989)

    Google Scholar 

  9. Kolmogorov, A.N.: Stationary sequences in Hilbert space. Bull. Math. Univ. Moscow2, 1–40 (1941)

    Google Scholar 

  10. Koosis, P.: Introduction to Hp Spaces. New York: Cambridge University Press 1980

    Google Scholar 

  11. Krein, M.G.: On a problem of extrapolation of A.N. Kolmogorov. Dokl. Akad. Nauk SSSR46, 306–309 (1945)

    Google Scholar 

  12. Makagon, A., Mandrekar, V.: The spectral representation of stable processes: Harmonizability and regularity. Probab. Theory Relat. Fields85, 1–11 (1990)

    Google Scholar 

  13. Miamee, A.G., Pourahmadi, M.: Wold decomposition, prediction and parameterization of stable processes with infinite variance. Probab. Theory Relat. Fields79, 145–164 (1988)

    Google Scholar 

  14. Rajput, B.S., Rama-Murthy, K.: Spectral representations of semi-stable processes, and semistable laws on Banach spaces. J. Multivariate Anal.21, 139–157 (1987)

    Google Scholar 

  15. Rajput, B.S., Rama-Murthy, K.: Spectral representations of complex semi-stable and other infinitely divisible stochastic processes. Stochastic Processes Appl.26, 141–159 (1987)

    Google Scholar 

  16. Rajput, B.S., Rosinski, J.: Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields82, 451–487 (1989)

    Google Scholar 

  17. Rajput, B.S., Rama-Murthy, K., Sundberg, C.: An extremal problem in Hp of the upper half plane with application to prediction of stochastic processes. In: Cambanis, S., Samorodnitsky, G., Taqqu, M.S. (ed.) Stable processes and related topics, pp. 191–252. Boston: Birkhäuser 1991

    Google Scholar 

  18. Schilder, M.: Some structure theorem for the symmetric stable laws. Ann. Math. Stat.41, 412–421 (1970)

    Google Scholar 

  19. Shapiro, H.S.: Topics in approximation theory (Lect. Notes Math., vol. 187, pp. 55–58) Berlin Heidelberg New York: Springer 1971

    Google Scholar 

  20. Singer, I.: Best approximation in normed linear space by elements of linear subspaces. Berlin Heidelberg New York: Springer 1970

    Google Scholar 

  21. Urbanik, K.: Random measures and harmonizable sequences. Stud. Math.31, 61–88 (1968)

    Google Scholar 

  22. Urbanik, K.: Prediction of strictly stationary sequences. Colloq. Math.12, 115–129 (1964)

    Google Scholar 

  23. Urbanik, K.: Some prediction problems for strictly stationary processes. Part I, Proc. Fifth Berkeley Symp. (Univ. California Press). Math. Stat. Probab.2, 235–258 (1967)

    Google Scholar 

  24. Weron, A.: Harmonizable stable processes on groups: spectral, ergodic and interpolation properties. Z. Wahrscheinlichkeitstheor. Verw. Geb.681, 473–491 (1985)

    Google Scholar 

  25. Wiener, N.: Extrapolation, interpolation and smoothing of stationary time series. New York: Wiley 1949

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Tennessee, 37996-1300, Knoxville, TN, USA

    Balram S. Rajput & Carl Sundberg

Authors
  1. Balram S. Rajput
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  2. Carl Sundberg
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Additional information

This research is partially supported by AFSOR Grant No. 90-016 8 and the University of Tennessee Science Alliance, a State of Tennessee Center of Excellence

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Rajput, B.S., Sundberg, C. On some extremal problems in Hp and the prediction ofL p-harmonizable stochastic processes. Probab. Th. Rel. Fields 99, 197–210 (1994). https://doi.org/10.1007/BF01199022

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  • Received: 26 November 1991

  • Revised: 21 December 1993

  • Issue Date: June 1994

  • DOI: https://doi.org/10.1007/BF01199022

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Mathematics Subject Classification

  • 30D55
  • 60G25
  • 62M20
  • 60E07
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