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Some strong laws of large numbers for blockwise martingale difference sequences in martingale type p Banach spaces

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Abstract

For a blockwise martingale difference sequence of random elements {V n , n ≥ 1} taking values in a real separable martingale type p (1 ≤ p ≤ 2) Banach space, conditions are provided for strong laws of large numbers of the form \(\lim _{n \to \infty } \sum\nolimits_{i = 1}^n {V_i /g_n = 0}\) almost surely to hold where the constants g n ↑ ∞. A result of Hall and Heyde [Martingale Limit Theory and Its Application, Academic Press, New York, 1980, p. 36] which was obtained for sequences of random variables is extended to a martingale type p (1 < p ≤ 2) Banach space setting and to hold with a Marcinkiewicz-Zygmund type normalization. Illustrative examples and counterexamples are provided.

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References

  1. Móricz, F.: Strong limit theorems for blockwise m-dependent and blockwise quasiorthogonal sequences of random variables. Proc. Amer. Math. Soc., 101, 709–715 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chow, Y. S., Teicher, H.: Probability Theory: Independence, Interchangeability, Martingales, Third Edition, Springer-Verlag, New York, 1997

    MATH  Google Scholar 

  3. Gaposhkin, V. F.: Series of block-orthogonal and block-independent systems. Izv. Vyssh. Uchebn. Zaved. Mat. 12–18 (1990) (in Russian); (English translation) Soviet Math. (Iz. VUZ Mat.), 34, 13–20 (1990)

  4. Gaposhkin, V. F.: On the strong law of large numbers for blockwise independent and blockwise orthogonal random variables. Teor. Veroyatnost. i Primenen., 39, 804–812 (1994) (in Russian); (English translation) Theory Probab. Appl., 39, 677–684 (1994)

    MathSciNet  MATH  Google Scholar 

  5. Hall, P., Heyde, C. C.: Martingale Limit Theory and Its Application, Academic Press, New York, 1980

    MATH  Google Scholar 

  6. Rosalsky, A., Thanh, L. V.: On the strong law of large numbers for sequences of blockwise independent and blockwise p-orthogonal random elements in Rademacher type p Banach spaces. Probab. Math. Statist., 27, 205–222 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Móricz, F., Su, K.-L., Taylor, R. L.: Strong laws of large numbers for arrays of orthogonal random elements in Banach spaces. Acta Math. Hungar., 65, 1–16 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Quang, N. V., Thanh, L. V.: On the strong laws of large numbers for two-dimensional arrays of blockwise independent and blockwise orthogonal random variables. Probab. Math. Statist., 25, 385–391 (2005)

    MathSciNet  MATH  Google Scholar 

  9. Thanh, L. V.: On the strong law of large numbers for d-dimensional arrays of random variables. Electron. Comm. Probab., 12, 434–441 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Móricz, F., Stadtmüller, U., Thalmaier, M.: Strong laws for blockwise M-dependent random fields. J. Theoret. Probab., 21, 660–671 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Stadtmüller, U., Thanh, L. V.: On the strong limit theorems for double arrays of blockwise M-dependent random variables. Acta Mathematica Sinica, English Series, 27(10), 1923–1934 (2011)

    Article  MathSciNet  Google Scholar 

  12. Scalora, F. S.: Abstract martingale convergence theorems. Pacific J. Math., 11, 347–374 (1961)

    MathSciNet  MATH  Google Scholar 

  13. Hoffmann-Jørgensen, J., Pisier, G.: The law of large numbers and the central limit theorem in Banach spaces. Ann. Probab., 4, 587–599 (1976)

    Article  Google Scholar 

  14. Pisier, G.: Martingales with values in uniformly convex spaces. Israel J. Math., 20, 326–350 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  15. Pisier, G.: Probabilistic methods in the geometry of Banach spaces. In: Probability and Analysis. Lectures given at the 1st 1985 Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held as Varenna (Como), Italy, 31 May–8 June, 1985 (Edited by G. Letta and M. Pratelli), Lecture Notes in Mathematics, Vol. 1206, Springer-Verlag, Berlin, 1986, 167–241

    Google Scholar 

  16. Rosalsky, A., Sreehari, M.: L r convergence theorems with or without random indices for randomly weighted sums of random elements in martingale type p Banach spaces. Stochastic Modelling Appl., 4, 1–15 (2001)

    Google Scholar 

  17. Woyczyński, W. A.: Geometry and martingales in Banach spaces. In: Probability — Winter School, Proceedings of the 4th Winter School on Probability, Karpacz, Poland, January 1975 (Edited by Z. Ciesielski, K. Urbanik, and W.A. Woyczyński), Lecture Notes in Mathematics, Vol. 472, Springer-Verlag, Berlin, 1975, 229–275

    Google Scholar 

  18. Woyczyński, W. A.: Asymptotic behavior of martingales in Banach spaces. In: Probability in Banach Spaces. Proceedings of the 1st International Conference on Probability in Banach Spaces, 20–26 July 1975, Oberwolfach (Edited by A. Beck), Lecture Notes in Mathematics, Vol. 526, Springer-Verlag, Berlin, 1976, 273–284

    Google Scholar 

  19. Schwartz, L.: Geometry and Probability in Banach Spaces, Notes by Paul R. Chernoff, Lecture Notes in Mathematics, Vol. 852, Springer-Verlag, Berlin, 1981

    MATH  Google Scholar 

  20. Taylor, R. L.: Stochastic Convergence of Weighted Sums of Random Elements in Linear Spaces, Lecture Notes in Mathematics, Vol. 672, Springer-Verlag, Berlin, 1978

    MATH  Google Scholar 

  21. Wei, D., Taylor, R. L.: Convergence of weighted sums of tight random elements. J. Multivariate Anal., 8, 282–294 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  22. Adler, A., Rosalsky, A., Taylor, R. L.: Some strong laws of large numbers for sums of random elements. Bull. Inst. Math. Acad. Sinica, 20, 335–357 (1992)

    MathSciNet  MATH  Google Scholar 

  23. Adler, A., Rosalsky, A.: Some general strong laws for weighted sums of stochastically dominated random variables. Stochastic Anal. Appl., 5, 1–16 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  24. Loève, M.: Probability Theory, Vol. I, Fourth Edition, Springer-Verlag, New York, 1977

    Google Scholar 

  25. Stout, W. F.: Almost Sure Convergence, Academic Press, New York, 1974

    MATH  Google Scholar 

  26. Adler, A., Rosalsky, A., Taylor, R. L.: Strong laws of large numbers for weighted sums of random elements in normed linear spaces. Internat. J. Math. & Math. Sci., 12, 507–529 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Statistics, University of Florida, Gainesville, Florida, 32611, USA

    Andrew Rosalsky

  2. Department of Mathematics, Vinh University, Nghe An, 42118, Vietnam

    Le Van Thanh

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  1. Andrew Rosalsky
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  2. Le Van Thanh
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Correspondence to Andrew Rosalsky.

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The second author is supported in part by the National Foundation for Science Technology Development, Vietnam (NAFOSTED) (Grant No. 101.02.32.09)

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Rosalsky, A., Van Thanh, L. Some strong laws of large numbers for blockwise martingale difference sequences in martingale type p Banach spaces. Acta. Math. Sin.-English Ser. 28, 1385–1400 (2012). https://doi.org/10.1007/s10114-012-0378-7

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  • DOI: https://doi.org/10.1007/s10114-012-0378-7

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