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Convergence of the series of large-deviation probabilities for sums of independent equally distributed random variables

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Abstract

The series\(\sum\nolimits_{n \geqslant 1} {\tau _n P(|S_n | \geqslant \varepsilon n^a )}\) is studied, where Sn are the sums of independent equally distributed random variables, τn is a sequence of nonnegative numbers, α>0, and ɛ>0 is an arbitrary positive number. For a broad class of sequences τn, the necessary and sufficient conditions are established for the convergence of this series for any ɛ>0.

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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 770–784, June, 1993.

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Klesov, O.I. Convergence of the series of large-deviation probabilities for sums of independent equally distributed random variables. Ukr Math J 45, 845–862 (1993). https://doi.org/10.1007/BF01061437

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Keywords

  • Broad Class
  • Nonnegative Number
  • Arbitrary Positive Number