Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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THE WEAK LAW OF LARGE NUMBERS FOR RANDOMLY WEIGHTED PARTIAL SUMS

  • Kim, Tae-Sung (Mathematical Science Division, Wonkwang University) ;
  • Choi, Kyu-Hyuck (Mathematical Science Division, Wonkwang University) ;
  • Lee, Il-Hyun (Mathematical Science Division, Wonkwang University)
  • Published : 1999.05.01
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Abstract

In this paper we establish the weak law of large numbers for randomly weighted partial sums of random variables and study conditions imposed on the triangular array of random weights {$W_{nj}{\;}:{\;}1{\leq}j{\leq}n,{\;}n{\geq}1$} and on the triangular array of random variables {$X_{nj}{\;}:{\;}1{\leq}j{\leq}n,{\;}{\geq}1$} which ensure that $\sum_{j=1}^{n}{\;}W_{nj}{\mid}X_{nj}{\;}-{\;}B_{nj}{\mid}$ converges In probability to 0, where {$B_{nj}{\;}:{\;}1{\;}{\leq}{\;}j{\;}{\leq}{\;}n,{\;}n{\;}{\geq}{\;}1$} is a centering array of constants or random variables.

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References

  1. J. Multivariate Anal. v.37 A weak law for normed weighted sums of random elements in Rademacher type p Banach spaces A. Adler;A. Rosalsky;R. L. Taylor
  2. Statist. Probab. Lett. v.26 Unconditional Glivenko-Cantelli type theroems and weak laws of large numbers for bootstrap E. Arenal-Gutierrez;C. Matran;J. A. Cuesta-Albertas
  3. Statist. Probab. Lett. v.1 Strong law for the bootstap K. B. Athreya
  4. Am. Probab v.1 Limiting behavior of weighted sums of independent random variables Y. S. Chow;T. L. Lai
  5. Statist. Probab. Lett. v.14 On the law of large numbers for the bootstrap mean Csorgo
  6. J. Theoretical Probab v.8 A strong law for weighted sums of random variables J. Cuzick
  7. Z. Wahrscheinlich keits theorie und verw. Gebiete v.4 Convergence of weighted averages of independent random variables B. Jamison;S. Orey;W. E. Prutt
  8. Theory of Point Estimation E. L. Lehmann
  9. J. Math. mech v.15 Summability of independent random variables W. E. Prutt
  10. Ann. Math. Statist v.39 Some results on the complete and almost sure convergence of linear combinations of independent random variables and martingale differences W. F. Stout
  11. Z. Wahrsch. theori und verw. Geb v.69 Almost certain convergence in double arrays H. Z. Teicher
  12. J. Japan Statist. Soc. v.20 On the uniform integrability and almost sure convergence of weighted sums K. F. Yu