Asymptotic behavior for sums of non-identically distributed random variables

  • Chang-jun Yu
  • Dong-ya ChengEmail author


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.


lower limits upper limits heavy-tailed distributions local distributions densities 

MR Subject Classification

Primary 60E05 60F99 


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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

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