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On Asymptotic Behavior of the Convolution of Distributions with Regularly Exponentially Decreasing Tails

In this note, we study the asymptotic behavior of tails of distributions and density of a sum of independent random variables in the case where the tails of distributions (densities) of the summands decrease exponentially at infinity.

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References

  1. 1.

    A. A. Borovkov, Asymptotic Analysis of Random Walks. Fast Decreasing Distributions of Increments [in Russian], Moscow (2013).

  2. 2.

    A. A. Borovkov and A. A. Mogulskii, “On large and superlarge deviations of sums of independent random vectors under the Cramér condition. II,” Teor. Veroyatn. Primen., 51, 641–673 (2007).

    MathSciNet  Article  Google Scholar 

  3. 3.

    W. Feller, Introduction to Probability Theory and its Applications, Vol. 2 [Russian translation], Moscow (1984).

  4. 4.

    D. B. H. Cline, “Convolution tails, product tails and domains of attraction,” Probab. Theory Relat. Fields, 72, 529–557 (1986).

    MathSciNet  Article  Google Scholar 

  5. 5.

    A. G. Pakes, “Convolution equivalence and infinite divisibility,” J. Appl. Probab., 41, 407–424 (2004).

    MathSciNet  Article  Google Scholar 

  6. 6.

    S. Zachary and S. G. Foss, “On the exact asymptotics of the maximum of a random walk with increments in a class of light-tailed distributions,” Sib. Mat. Zh., 47, 1034–1041 (2006).

    MathSciNet  Article  Google Scholar 

  7. 7.

    S. G. Foss, “On the exact asymptotics of the stationary distribution of sojourn time in a tandem of queuing systems for a class of light-tailed distributions,” Probl. Pered. Inform., 43, 93–108 (2007).

    Google Scholar 

  8. 8.

    T. Watanabe, “Convolution equivalence and distributions of random sums,” Probab. Theory Relat. Fields, 142, 367–397 (2008).

    MathSciNet  Article  Google Scholar 

  9. 9.

    A. A. Borovkov, Probability Theory [in Russian], Moscow (1999).

  10. 10.

    L. V. Rozovsky, “On superlarge deviations of a sum of independent random variables with a common absolutely continuous distribution satisfying the Cramér condition,” Teor. Veroyatn. Primen., 48, 78–103 (2003).

    Article  Google Scholar 

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Correspondence to L. V. Rozovsky.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 486, 2019, pp. 265–274.

Translated by L. V. Rozovsky.

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Rozovsky, L.V. On Asymptotic Behavior of the Convolution of Distributions with Regularly Exponentially Decreasing Tails. J Math Sci 258, 920–926 (2021). https://doi.org/10.1007/s10958-021-05591-0

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