Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON THE PRUSS EXTENSION OF THE HSU-ROBBINS-ERD S THEOREM

  • Sung, Soo-Hak (Department of Applied Mathematics, Pai Chai University)
  • Published : 1999.05.01
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Abstract

The Hsu-Robbins-erd s theorem states that if {$X_m,n\geq1$} is a sequence of independent and identically distributed random variables, then ${EX_1}^2<\infty$ and $EX_1$=0 if and only if ${\sum_{n=1}}^\infty\;P($\mid${\sum_{k=1}}^nX_k$\mid$\geqn\in)<\infty$ for every $\in$ > 0. Under some auxiliary conditions, Sp taru (1994) extended this to the case where the $X_n$ are independent, but their distributions come from a finite set. Pruss (1996) proved Sp taru's result under weaker conditions, The purpose of this paper is to improve Pruss conditions.

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References

  1. Ann. Math. Statist v.20 On a theorem of Hsu and Robbins P. Erdos
  2. Ann. Math. Statist v.21 Remark on my paper "On a theorem of Hsu and Robbins P. Erdos
  3. Proc. Nat. Acad. Sci. v.33 Complete convergence and the law of large numbers P. L. Hsu;H. Robbins
  4. J. Math. Anal. Appl. v.199 On Spataru's extension of the Hsu-Robbins-Erdos law of large numbers A. R. Pruss
  5. J. Math. Anal. Appl. v.187 A generalization of the Hsu-Robbins-Erdos theorem A. Spataru
  6. 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
  7. J. Math. Anal. Appl. v.206 A note on Spataru's complete convergence theorem S. H. Sung