Abstract and Applied Analysis

Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences

Shunli Hao

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Abstract

We study the convergence rates in the law of large numbers for arrays of Banach valued martingale differences. Under a simple moment condition, we show sufficient conditions about the complete convergence for arrays of Banach valued martingale differences; we also give a criterion about the convergence for arrays of Banach valued martingale differences. In the special case where the array of Banach valued martingale differences is the sequence of independent and identically distributed real valued random variables, our result contains the theorems of Hsu-Robbins-Erdös (1947, 1949, and 1950), Spitzer (1956), and Baum and Katz (1965). In the real valued single martingale case, it generalizes the results of Alsmeyer (1990). The consideration of Banach valued martingale arrays (rather than a Banach valued single martingale) makes the results very adapted in the study of weighted sums of identically distributed Banach valued random variables, for which we prove new theorems about the rates of convergence in the law of large numbers. The results are established in a more general setting for sums of infinite many Banach valued martingale differences. The obtained results improve and extend those of Ghosal and Chandra (1998).

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 715054, 26 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393512156

Digital Object Identifier
doi:10.1155/2013/715054

Mathematical Reviews number (MathSciNet)
MR3132544

Zentralblatt MATH identifier
07095265

Citation

Hao, Shunli. Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences. Abstr. Appl. Anal. 2013 (2013), Article ID 715054, 26 pages. doi:10.1155/2013/715054. https://projecteuclid.org/euclid.aaa/1393512156


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References

  • P. L. Hsu and H. Robbins, “Complete convergence and the law of large numbers,” Proceedings of the National Academy of Sciences of the United States of America, vol. 33, pp. 25–31, 1947.
  • P. Erdös, “On a theorem of Hsu and Robbins,” Annals of Mathematical Statistics, vol. 20, pp. 286–291, 1949.
  • P. Erdös, “Remark on my paper `On a theorem of Hsu and Robbin',” Annals of Mathematical Statistics, vol. 21, article 138, 1950.
  • F. Spitzer, “A combinatorial lemma and its application to probability theory,” Transactions of the American Mathematical Society, vol. 82, pp. 323–339, 1956.
  • M. L. Katz, “The probability in the tail of a distribution,” Annals of Mathematical Statistics, vol. 34, pp. 312–318, 1963.
  • L. E. Baum and M. Katz, “Convergence rates in the law of large numbers,” Transactions of the American Mathematical Society, vol. 120, pp. 108–123, 1965.
  • T. L. Lai, “Limit theorems for delayed sums,” Annals of Probability, vol. 2, pp. 432–440, 1974.
  • M. U. Gafurov and A. D. Slastnikov, “Some problems on the exit of a random walk from a curvilinear boundary, and large deviations,” Theory of Probability and Its Applications, vol. 32, no. 2, pp. 327–348, 1987.
  • D. L. Li, M. B. Rao, T. F. Jiang, and X. C. Wang, “Complete convergence and almost sure convergence of weighted sums of random variables,” Journal of Theoretical Probability, vol. 8, no. 1, pp. 49–76, 1995.
  • T.-C. Hu, D. Szynal, and A. I. Volodin, “A note on complete convergence for arrays,” Statistics & Probability Letters, vol. 38, no. 1, pp. 27–31, 1998.
  • T.-C. Hu and A. Volodin, “Addendum to: `A note on complete convergence for arrays`,” Statistics & Probability Letters, vol. 38, no. 1, pp. 27–31, 1998.
  • T.-C. Hu, M. Ordóñez Cabrera, S. H. Sung, and A. Volodin, “Complete convergence for arrays of rowwise independent random variables,” Korean Mathematical Society, vol. 18, no. 2, pp. 375–383, 2003.
  • A. Kuczmaszewska, “On some conditions for complete convergence for arrays,” Statistics & Probability Letters, vol. 66, no. 4, pp. 399–405, 2004.
  • S. H. Sung, A. I. Volodin, and T.-C. Hu, “More on complete convergence for arrays,” Statistics & Probability Letters, vol. 71, no. 4, pp. 303–311, 2005.
  • V. M. Kruglov, A. I. Volodin, and T.-C. Hu, “On complete convergence for arrays,” Statistics & Probability Letters, vol. 76, no. 15, pp. 1631–1640, 2006.
  • E. Lesigne and D. Volný, “Large deviations for martingales,” Stochastic Processes and their Applications, vol. 96, no. 1, pp. 143–159, 2001.
  • G. Stoica, “Baum-Katz-Nagaev type results for martingales,” Journal of Mathematical Analysis and Applications, vol. 336, no. 2, pp. 1489–1492, 2007.
  • G. Alsmeyer, “Convergence rates in the law of large numbers for martingales,” Stochastic Processes and their Applications, vol. 36, no. 2, pp. 181–194, 1990.
  • S. Ghosal and T. K. Chandra, “Complete convergence of martingale arrays,” Journal of Theoretical Probability, vol. 11, no. 3, pp. 621–631, 1998.
  • A. Gut, “Complete convergence and Cesàro summation for i.i.d. random variables,” Probability Theory and Related Fields, vol. 97, no. 1-2, pp. 169–178, 1993.
  • H. Lanzinger and U. Stadtmüller, “Baum-Katz laws for certain weighted sums of independent and identically distributed random variables,” Bernoulli, vol. 9, no. 6, pp. 985–1002, 2003.
  • Y. Wang, X. Liu, and C. Su, “Equivalent conditions of complete convergence for independent weighted sums,” Science in China A, vol. 41, no. 9, pp. 939–949, 1998.
  • K. F. Yu, “Complete convergence of weighted sums of martingale differences,” Journal of Theoretical Probability, vol. 3, no. 2, pp. 339–347, 1990.
  • D. L. Li, M. B. Rao, and X. C. Wang, “Complete convergence of moving average processes,” Statistics & Probability Letters, vol. 14, no. 2, pp. 111–114, 1992.
  • Q. M. Shao, “Complete convergence for $\alpha $-mixing sequences,” Statistics & Probability Letters, vol. 16, no. 4, pp. 279–287, 1993.
  • Q. M. Shao, “Maximal inequalities for partial sums of $\rho $-mixing sequences,” Annals of Probability, vol. 23, no. 2, pp. 948–965, 1995.
  • Z. Szewczak, “On Marcinkiewicz-Zygmund laws,” Journal of Mathematical Analysis and Applications, vol. 375, no. 2, pp. 738–744, 2011.
  • J.-I. Baek and S.-T. Park, “Convergence of weighted sums for arrays of negatively dependent random variables and its applications,” Journal of Theoretical Probability, vol. 23, no. 2, pp. 362–377, 2010.
  • H.-Y. Liang, “Complete convergence for weighted sums of negatively associated random variables,” Statistics & Probability Letters, vol. 48, no. 4, pp. 317–325, 2000.
  • H.-Y. Liang and C. Su, “Complete convergence for weighted sums of NA sequences,” Statistics & Probability Letters, vol. 45, no. 1, pp. 85–95, 1999.
  • A. Kuczmaszewska, “On complete convergence in Marcinkiewicz-Zygmund type SLLN for negatively associated random variables,” Acta Mathematica Hungarica, vol. 128, no. 1-2, pp. 116–130, 2010.
  • V. M. Kruglov, “Complete convergence for maximal sums of negatively associated random variables,” Journal of Probability and Statistics, vol. 2010, Article ID 764043, 17 pages, 2010.
  • M.-H. Ko, “On the complete convergence for negatively associated random fields,” Taiwanese Journal of Mathematics, vol. 15, no. 1, pp. 171–179, 2011.
  • Y. S. Chow and H. Teicher, Probability Theory: Independent, Interchangeability, Martingales, Springer, New York, NY, USA, 3rd edition, 1997.
  • J. Baxter, R. Jones, M. Lin, and J. Olsen, “SLLN for weighted independent identically distributed random variables,” Journal of Theoretical Probability, vol. 17, no. 1, pp. 165–181, 2004.
  • G. Pisier, “Martingales with values in uniformly convex spaces,” Israel Journal of Mathematics, vol. 20, no. 3-4, pp. 326–350, 1975.
  • N. V. Huan and N. V. Quang, “The Doob inequality and strong law of large numbers for multidimensional arrays in general Banach spaces,” Kybernetika, vol. 48, no. 2, pp. 254–267, 2012.
  • P. Assouad, “Espaces p-lisses et q-convexes, inégalités de Burkholder,” in Proceedings of the Séminaire Maurey-Schwartz 1974–1975, Espaces L$^{p}$, Applications Radonifiantes et Géométrie des Espaces de Banach, no. 15, p. 8, Centre de Mathématiques Appliquées-École Polytechnique, 1975.
  • N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, vol. 27, Cambridge University Press, Cambridge, UK, 1987.
  • R. Salem and A. Zygmund, “Some properties of trigonometric series whose terms have random signs,” Acta Mathematica, vol. 91, pp. 245–301, 1954.
  • J. D. Hill, “The Borel property of summability methods,” Pacific Journal of Mathematics, vol. 1, pp. 399–409, 1951.
  • D. L. Hanson and L. H. Koopmans, “On the convergence rate of the law of large numbers for linear combinations of independent random variables,” Annals of Mathematical Statistics, vol. 36, pp. 559–564, 1965.
  • W. E. Pruitt, “Summability of independent random variables,” Journal of Applied Mathematics and Mechanics, vol. 15, pp. 769–776, 1966.
  • W. E. Franck and D. L. Hanson, “Some results giving rates of convergence in the law of large numbers for weighted sums of independent random variables,” Bulletin of the American Mathematical Society, vol. 72, pp. 266–268, 1966.
  • Y. S. Chow, “Some convergence theorems for independent random variables,” Annals of Mathematical Statistics, vol. 37, pp. 1482–1493, 1966.
  • Y. S. Chow and T. L. Lai, “Limiting behavior of weighted sums of independent random variables,” Annals of Probability, vol. 1, pp. 810–824, 1973.
  • W. F. Stout, “Some results on the complete and almost sure convergence of linear combinations of independent random variables and martingale differences,” Annals of Mathematical Statistics, vol. 39, pp. 1549–1562, 1968. \endinput