Journal of the Korean Statistical Society

The Korean Statistical Society (한국통계학회)
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On the Almost Certain Rate of Convergence of Series of Independent Random Variables

  • Nam, Eun-Woo (Department of Computer Science and Statistics, Air Force Academy, Cheongjoo 363-849) ;
  • Andrew Rosalsky (Department of Statistics, University of Florida, Gainesville, Florida 32611, USA)
  • Published : 1995.06.01
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Abstract

The rate of convergence to a random variable S for an almost certainly convergent series $S_n = \sum^n_{j=1} X_j$ of independent random variables is studied in this paper. More specifically, when $S_n$ converges to S almost certainly, the tail series $T_n = \sum^{\infty}_{j=n} X_j$ is a well-defined sequence of random variable with $T_n \to 0$ a.c. Various sets of conditions are provided so that for a given numerical sequence $0 < b_n = o(1)$, the tail series strong law of large numbers $b^{-1}_n T_n \to 0$ a.c. holds. Moreover, these results are specialized to the case of the weighted i.i.d. random varialbes. Finally, example are provided and an open problem is posed.

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References

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