The laws of large numbers for Pareto-type random variables under sub-linear expectation

Abstract

In this paper, some laws of large numbers are established for random variables that satisfy the Pareto distribution, so that the relevant conclusions in the traditional probability space are extended to the sub-linear expectation space. Based on the Pareto distribution, we obtain the weak law of large numbers and strong law of large numbers of the weighted sum of some independent random variable sequences.

Keywords

Sub-linear expectation / Pareto type distribution / laws of large numbers / independent and identical distribution

References

[1] Adler A. Laws of large numbers for two tailed Pareto random variables. Probab Math Statist 2008; 28(1): 121–128

[2] Adler A. An exact weak law of large numbers. Bull Inst Math Acad Sin (N S) 2012; 7(3): 417–422

[3] ChenJ. Limit theory for sub-linear expectations and its applications. Ph D Thesis, Jinan: Shandong University, 2014 (in Chinese)

[4] Chen Z J. Strong laws of large numbers for sub-linear expectations. Sci China Math 2016; 59(5): 945–954

[5] Chen Z J, Wu P Y, Li B M. A strong law of large numbers for non-additive probabilities. Internat J Approx Reason 2013; 54(3): 365–377

[6] Hu C. A strong law of large numbers for sub- linear expectation under a general moment condition. Statist Probab Lett 2016; 119: 248–258

[7] Hu C. Weak and strong laws of large numbers for sub-linear expectation. Comm Statist Theory Methods 2020; 49(2): 430–440

[8] MaX CWu Q Y. On some conditions for strong law of large numbers for weighted sums of END random variables under sub-linear expectations. Discrete Dyn Nat Soc, 2019: 7945431 (8 pp)

[9] Marinacci M. Limit laws for non-additive probabilities and their frequentist interpretation. J Econom Theory 1999; 84(2): 145–195

[10] PengS G. G-expectation, G-Brownian motion and related stochastic calculus of Itô type, In: Stoch Anal Appl, Abel Symp, Vol 2, Berlin: Springer, 2007: 541–567

[11] Peng S G. Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stochastic Process Appl 2008; 118(12): 2223–2253

[12] Peng S G. Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sub-linear expectations. Sci China Ser A 2009; 52(7): 1391–1411

[13] PengS G. Law of large numbers and central limit theorem under nonlinear expectations. Probab Uncertain Quant Risk, 2019, 4: 4 (8 pp)

[14] Wu Q Y, Jiang Y Y. Strong law of large numbers and Chover’s law of the iterated logarithm under sub-linear expectations. J Math Anal Appl 2018; 460(1): 252–270

[15] Yang W Z, Yang L, Wei D, Hu S H. The laws of large numbers for Pareto-type random variables with infinite means. Comm Statist Theory Methods 2019; 48(12): 3044–3054

[16] Zhang L X. Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications. Sci China Math 2016; 59(4): 751–768

[17] Zhang L X. Strong limit theorems for extended independent random variables and extended negatively dependent random variables under sub-linear expectations. Acta Math Sci 2022; 42: 467–490

[18] Zhang L X. Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm. Sci China Math 2016; 59(12): 2503–2526

RIGHTS & PERMISSIONS

Higher Education Press 2022