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Order law of large numbers of the Marcinkiewicz–Zygmund type

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The order law of large numbers of the Marcinkiewicz–Zygmund type is established for random variables on Banach lattices. Similar results are also obtained for the maximum scheme.

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References

  1. 1.

    J. Marcinkiewicz and A. Zygmund, “Sur les fonctions indépendantes,” Fund. Math., 29, 60–90 (1937).

  2. 2.

    M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer, Berlin (1991).

  3. 3.

    I. K. Matsak and A. M. Plichko, “On the Marcinkiewicz–Zygmund law of large numbers in Banach lattices,” Ukr. Mat. Zh., 62, No. 4, 504–513 (2010); English translation: Ukr. Math. J., 62, No. 4, 575–587 (2010).

  4. 4.

    J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Springer, Berlin (1979).

  5. 5.

    L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1984).

  6. 6.

    I. K. Matsak, “A remark on the order law of large numbers,” Theor. Imovir. Mat. Stat., Issue 72, 84–92 (2005); English translation: Theory Probab. Math. Statist., No. 72, 93–102 (2006).

  7. 7.

    A.V. Bukhvalov, A. I. Veksler, and V. A. Geiler, “Normed lattices,” in: VINITI Series in Mathematical Analysis [in Russian], Issue 18, VINITI, Moscow (1980), pp. 125–184.

  8. 8.

    W. Feller, An Introduction to Probability Theory and Its Applications. Vol. 2 [Russian translation], Mir, Moscow (1984).

  9. 9.

    I. K. Matsak and A. M. Plichko, “On the maxima of independent random elements in Banach functional lattices,” Theor. Imovir. Mat. Stat., Issue 61, 105–116 (1999); English translation: Theory Probab. Math. Statist., No. 61, 109–120 (2000).

  10. 10.

    J. A. Wellner, “A martingale inequality for the empirical process,” Ann. Probab., No. 2, 303–308 (1977).

  11. 11.

    N. N. Vakhaniya, V. I. Tarieladze, and S. A. Chobanyan, Probability Distributions in Banach Spaces [in Russian], Nauka, Moscow (1985).

  12. 12.

    K. Yosida, Functional Analysis [Russian translation], Mir, Moscow (1967).

  13. 13.

    I. K. Matsak, “Evaluation of the moments of suprema of normalized sums of independent random variables,” Theor. Imovirn. Mat. Stat., Issue 67, 104–116 (2002); English translation: Theory Probab. Math. Statist., No. 67, 115–128 (2003).

  14. 14.

    A.V. Skorokhod, Random Processes with Independent Increments [in Russian], Nauka, Moscow (1964).

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Author information

Correspondence to K. S. Akbash.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 12, pp. 1587–1597, December, 2010.

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Akbash, K.S., Matsak, I.K. Order law of large numbers of the Marcinkiewicz–Zygmund type. Ukr Math J 62, 1839–1851 (2011). https://doi.org/10.1007/s11253-011-0474-3

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Keywords

  • English Translation
  • Independent Random Variable
  • Symmetric Case
  • Banach Lattice
  • Random Element