Journal of Theoretical Probability

, Volume 9, Issue 3, pp 797–809 | Cite as

The strong law of large numbers for weighted averages under dependence assumptions

  • Tapas K. Chandra
  • Subhashis Ghosal
Article

Abstract

Strong laws of large numbers (SLLN) for weighted averages are proved under various dependence assumptions when the variables are not necessarily independent or identically distributed. The results considerably extend the existing results. Weighted versions of the Marcinkiewicz-Zygmund SLLN are also formulated and proved under a similar set up. It seems that such results are not known even for independent and identically distributed random variables.

Key Words

Asymptotically quadrant sub-independence Marcinkiewicz-Zygmund strong laws maximal inequalities mixing conditions mixingale difference strong laws weighted averages 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Birkel, T. (1992). Laws of large numbers under dependence assumptions.Statist. Prob. Lett. 14, 355–362.Google Scholar
  2. 2.
    Chandra, T. K. (1991). Extensions of Rajchman's strong law of large numbers.Sankhyā, Ser. A 53, 118–121.Google Scholar
  3. 3.
    Chandra, T. K., and Ghosal, S. (1993). Some elementary strong laws of large numbers: a review. Technical Report #22/93, Indian Statistical Institute.Google Scholar
  4. 4.
    Chandra, T. K., and Ghosal, S. (1996). Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund.Acta Math. Hung. 72(3) (to appear).Google Scholar
  5. 5.
    Etemadi, N. (1983). Stability of sums of weighted random variables.J. Multivariate Anal. 13, 361–365.Google Scholar
  6. 6.
    Hall, P., and Heyde, C. C. (1980).Martingale Limit Theory and Its Application., Academic Press, New York.Google Scholar
  7. 7.
    Jamison, B., Orey, S., and Pruitt, W. E. (1965). Convergence of weighted averages of independent random variables.Z. Wahrsch. Verw. Gebiete 4, 40–44.Google Scholar
  8. 8.
    McLeish, D. L. (1975). A maximal inequality and dependent strong laws.Ann. Prob. 3, 829–839.Google Scholar
  9. 9.
    Pruitt, W. E. (1966). Summability of independent random variables.J. Math. Mech. 15, 769–776.Google Scholar
  10. 10.
    Rothatgi, V. K. (1971). Convergence of weighted sums of independent random variables.Proc. Cambridge Phil. Soc. 69, 305–307.Google Scholar
  11. 11.
    Rosalsky, A. (1987). Strong stability of normed sums of pairwise i.i.d. random variables.Bull. Inst. Math. Acad. Sinica 15, 203–219.Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • Tapas K. Chandra
    • 1
  • Subhashis Ghosal
    • 1
  1. 1.Division of Theoretical Statistics and MathematicsIndian Statistical InstituteCalcuttaIndia

Personalised recommendations