Advertisement

Asymptotic behavior for sums of non-identically distributed random variables

  • Chang-jun Yu
  • Dong-ya ChengEmail author
Article
  • 6 Downloads

Abstract

For any given positive integer m, let Xi, 1 ≤ im be m independent random variables with distributions Fi, 1 ≤ im. When all the summands are nonnegative and at least one of them is heavy-tailed, we prove that the lower limit of the ratio \(\frac{{P(\sum\nolimits_{i = 1}^m {{X_i}} > x})}{{\sum\nolimits_{i = 1}^m {{{\overline F }_i}(x)} }}\) equals 1 as x → ∞. When the summands are real-valued, we also obtain some asymptotic results for the tail probability of the sums. Besides, a local version as well as a density version of the above results is also presented.

Keywords

lower limits upper limits heavy-tailed distributions local distributions densities 

MR Subject Classification

Primary 60E05 60F99 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D Denisov, S Foss, D Korshunov. On lower limits and equivalence for distribution tails of random stopped sums, Bernoulli, 2008a, 14: 391–404.zbMATHGoogle Scholar
  2. [2]
    D Denisov, S Foss, D Korshunov. Lower limits for distribution tails of randomly stopped sums, Theor Probab Appl, 2008b, 52: 690–699.CrossRefzbMATHGoogle Scholar
  3. [3]
    S Foss, D Korshunov. Lower limits and equivalences for convolution tails, Ann Probab, 2007, 35: 366–383.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    S Foss, D Korshunov, S Zachary. An introduction to heavy-tailed and subexponential distributions, 2013, 2nd ed, New York, Springer.CrossRefzbMATHGoogle Scholar
  5. [5]
    S I Resnick. Heavy-tail phenomena: probabilistic and statistical modeling, 2007, Springer.zbMATHGoogle Scholar
  6. [6]
    W Rudin. Limits of ratios of tails of measures, Ann Probab, 1973, 1: 982–994.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Z Su, CSu, Z Hu, J Liu. On domination problem of non-negative distributions, Front Math China, 2009, 4: 681–696.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    T Watanabe, K Yamamuro. Local subexponentiality and self-decomposability, J Theor Probab, 2010, 23: 1039–1067.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    C Yu, Y Wang, Z Cui. Lower limits and upper limits for tails of random sums supported on R, Statist Probab Lett, 2010, 80: 1111–1120.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    KC Yuen, C Yin. Asymptotic results for tail probabilities of sums of dependent and heavy-tailed random variables, Chinese Annals of Mathematics, Series B, 2012, 33: 557–568.zbMATHGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics 2019

Authors and Affiliations

  1. 1.School of SciencesNantong UniversityNantongChina
  2. 2.School of Mathematical SciencesSoochow UniversitySuzhouChina
  3. 3.The Statistics and Operations Research Departmentthe University of North CarolinaChapel HillUSA

Personalised recommendations