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Amarts on Riesz spaces

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Abstract

The concepts of conditional expectations, martingales and stopping times were extended to the Riesz space context by Kuo, Labuschagne and Watson (Discrete time stochastic processes on Riesz spaces, Indag. Math., 15 (2004), 435–451). Here we extend the definition of an asymptotic martingale (amart) to the Riesz spaces context, and prove that Riesz space amarts can be decomposed into the sum of a martingale and an adapted sequence convergent to zero. Consequently an amart convergence theorem is deduced.

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References

  1. Edgar, G. A., Sucheston, L.: Amarts: A class of asymptotic martingales, Parts A and B. J. Multivariate Anal., 6, 193–221, 572–591 (1976)

    Article  MathSciNet  Google Scholar 

  2. Edgar, G. A., Sucheston, L.: Martingales in the limit and amarts. Proc. Amer. Math. Soc., 67, 315–320 (1977)

    Article  MathSciNet  Google Scholar 

  3. Edgar, G. A., Sucheston, L.: Stopping times and directed processes, Cambridge, Cambridge University Press, Cambridge, The United Kingdom, 1992

    MATH  Google Scholar 

  4. Egghe, L.: Stopping time techniques for analysts and probabilists, Cambridge University Press, 1984

  5. Ghoussoub, N.: Orderamarts: A class of asymptotic martingales. J. Multivariate Anal., 9, 165–172 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  6. Mucci, A. G.: Limits for martingale-like sequences. Pacific J. Math., 48, 197–202 (1973)

    MATH  MathSciNet  Google Scholar 

  7. Mucci, A. G.: Another martingale convergence theorem. Pacific J. Math., 64, 539–541 (1976)

    MATH  MathSciNet  Google Scholar 

  8. Wang, Z. P.: Convergence of vector-valued pramarts indexed by directed sets. Chinese Ann. Math. Ser. A, 10, 345–350 (1989)

    MATH  MathSciNet  Google Scholar 

  9. Xue, X. H.: T-uniform amarts and almost sure convergence in Banach space. Acta Math. Sci. (English Ed.), 5, 85–91 (1985)

    MATH  MathSciNet  Google Scholar 

  10. Xue, X. H.: Convergence of pramarts in Banach space. Acta Mathematica Sinica, Chinese Series, 29, 389–392 (1986)

    MATH  Google Scholar 

  11. Diestel, J., Jr. Uhl, J. J.: Vector measures, American Mathematical Society, 1977

  12. Dodds, P. G., Huismans, C. B., de Pagter, B.: Characterizations of conditional expectation-type operators. Pacific J. Math., 141, 55–77 (1990)

    MATH  MathSciNet  Google Scholar 

  13. Grobler, J. J., de Pagter, B.: Operators representable as multiplication-conditional expectation operators. J. Operator Theory, 48, 15–40 (2002)

    MATH  MathSciNet  Google Scholar 

  14. Karatzas, I., Shreve, S. E.: Brownian motion and stochastic processes, Springer Verlag, Heidelberg, Germany, 1991

    Google Scholar 

  15. Neveu, J.: Discrete-parameter martingales, North-Holland Publishing Co., Amsterdam, The Netherlands, 1975

    MATH  Google Scholar 

  16. Rao, M. M.: Foundations of stochastic processes, Academic Press, 1981

  17. Ghoussoub, N.: Summability and vector amarts. J. Multivariate Anal., 9, 173–178 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ghoussoub, N.: Riesz-space-valued measures and processes. Bull. Soc. Math. France, 110, 233–257 (1982)

    MATH  MathSciNet  Google Scholar 

  19. Ghoussoub, N., Sucheston, L.: A refinement of the Riesz decomposition for amarts amd semiamarts. J. Multivariate Anal., 8, 146–150 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kuo, W. C., Labuschagne, C. C. A., Watson, B. A.: Discrete time stochastic processes on Riesz spaces. Indag. Math., 15, 435–451 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kuo, W. C., Labuschagne, C. C. A., Watson, B. A.: An upcrossing theorem for martingales on Riesz spaces. Soft methodology and random information systems, Springer Verlag, Heidelberg, Germany, 2004, 101–108

    Google Scholar 

  22. Kuo, W. C., Labuschagne, C. C. A., Watson, B. A.: Conditional expectations on Riesz spaces, J. Math. Anal. Appl. 303, 509–521 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kuo, W. C., Labuschagne, C. C. A., Watson, B. A.: Convergence of Riesz space martingales. Indag. Math., in press

  24. Stoica, G.: On some stochastic-type operators. Analele Universitătii Bucuresti, Mathematică, 39, 58–62 (1990)

    MathSciNet  Google Scholar 

  25. Stoica, G.: Vector valued quasi-martingales. Stud. Cerc. Mat., 42, 73–79 (1990)

    MATH  MathSciNet  Google Scholar 

  26. Stoica, G.: The structure of stochastic processes in normed vector lattices. Stud. Cerc. Mat., 46, 477–486 (1994)

    MATH  MathSciNet  Google Scholar 

  27. Zaanen, A. C.: Introduction to Operator Theory in Riesz Space, Springer Verlag, Heidelberg, Germany, 1997

    Google Scholar 

  28. Meyer-Nieberg, P.: Banach lattices, Springer-Verlag, Heidelberg, Germany, 1991

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, University of the Witwatersrand, Private Bag 3, P O WITS 2050, Witwatersrand, South Africa

    Wen Chi Kuo, Coenraad C. A. Labuschagne & Bruce A. Watson

Authors
  1. Wen Chi Kuo
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  2. Coenraad C. A. Labuschagne
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  3. Bruce A. Watson
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Correspondence to Wen Chi Kuo.

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Supported in part by the John Knopfmacher Centre for Applicable Analysis and Number Theory

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Kuo, W.C., Labuschagne, C.C.A. & Watson, B.A. Amarts on Riesz spaces. Acta. Math. Sin.-English Ser. 24, 329–342 (2008). https://doi.org/10.1007/s10114-007-1025-6

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  • DOI: https://doi.org/10.1007/s10114-007-1025-6

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