Abstract
For any given positive integer m, let Xi, 1 ≤ i ≤ m be m independent random variables with distributions Fi, 1 ≤ i ≤ m. 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.
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References
D Denisov, S Foss, D Korshunov. On lower limits and equivalence for distribution tails of random stopped sums, Bernoulli, 2008a, 14: 391–404.
D Denisov, S Foss, D Korshunov. Lower limits for distribution tails of randomly stopped sums, Theor Probab Appl, 2008b, 52: 690–699.
S Foss, D Korshunov. Lower limits and equivalences for convolution tails, Ann Probab, 2007, 35: 366–383.
S Foss, D Korshunov, S Zachary. An introduction to heavy-tailed and subexponential distributions, 2013, 2nd ed, New York, Springer.
S I Resnick. Heavy-tail phenomena: probabilistic and statistical modeling, 2007, Springer.
W Rudin. Limits of ratios of tails of measures, Ann Probab, 1973, 1: 982–994.
Z Su, CSu, Z Hu, J Liu. On domination problem of non-negative distributions, Front Math China, 2009, 4: 681–696.
T Watanabe, K Yamamuro. Local subexponentiality and self-decomposability, J Theor Probab, 2010, 23: 1039–1067.
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.
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.
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Supported by the National Natural Science Foundation of China (no.11401415),Tian Yuan Foundation (nos.11226208 and 11426139), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no.13KJB110025), Postdoctoral Research Program of Jiangsu Province of China (no.1402111C) and Jiangsu Overseas Research and Training Program for Prominent University Young and Middle-aged Teachers and Presidents.
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Yu, Cj., Cheng, Dy. Asymptotic behavior for sums of non-identically distributed random variables. Appl. Math. J. Chin. Univ. 34, 45–54 (2019). https://doi.org/10.1007/s11766-019-3440-8
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DOI: https://doi.org/10.1007/s11766-019-3440-8