March  2022, 7(1): 1-12. doi: 10.3934/puqr.2022001

On the laws of the iterated logarithm with mean-uncertainty under sublinear expectations

1. 

Research Center for Mathematics and Interdisciplinary Sciences, Shandong University,Qingdao 266237, Shandong, China

2. 

Frontiers Science Center for Nonlinear Expectations (Ministry of Education), Shandong University, Qingdao 266237, Shandong, China

3. 

School of Mathematics, Shandong University, Jinan 250100, Shandong, China

lixinpeng@sdu.edu.cn

Received  January 14, 2022 Accepted  March 06, 2022 Published  March 2022 Early access  March 2022

Fund Project: The authors thank Professor Li-Xin Zhang for the constructive discussion concerning the relationship between regularity of the sublinear expectation and the existence of countably sub-additive capacity. This work is supported by NSF of Shandong Province (Grant No.ZR2021MA018), National Key R&D Program of China (Grant No.2018YFA0703900), NSF of China (Grant No.11601281) and the Young Scholars Program of Shandong University.

A new Hartman–Wintner-type law of the iterated logarithm for independent random variables with mean-uncertainty under sublinear expectations is established by the martingale analogue of the Kolmogorov law of the iterated logarithm in classical probability theory.

Citation: Xiaofan Guo, Shan Li, Xinpeng Li. On the laws of the iterated logarithm with mean-uncertainty under sublinear expectations. Probability, Uncertainty and Quantitative Risk, 2022, 7 (1) : 1-12. doi: 10.3934/puqr.2022001
References:
[1]

Chen, Z. and Hu, F., A law of the iterated logarithm under sublinear expectations, Journal of Financial Engineering, 2014, 1(2): 1450015. doi: 10.1142/S2345768614500159.

[2]

Feller, W., The general form of the so-called law of the iterated logarithm, Transactions of the American Mathematical Society, 1943, 54(3): 373−402. doi: 10.1090/S0002-9947-1943-0009263-7.

[3]

Hu, M., Li, X. and Li, X., Convergence rate of Peng’s law of large numbers under sublinear expectations, Probab. Uncertain. Quant. Risk, 2021, 6(3): 261−266. doi: 10.3934/puqr.2021013.

[4]

Guo, X., Li, S. and Li, X., On the Hartman-Wintner law of the iterated logarithm under sublinear expectation, Communications in Statistics-Theory and Methods, 2022, https://doi.org/10.1080/03610926.2022.2026394.

[5]

Guo, X. and Li, X., On the laws of large numbers for pseudo-independent random variables under sublinear expectation, Statistic and Probability Letters, 2021, 172: 109042. doi: 10.1016/j.spl.2021.109042.

[6]

Hartman, P. and Wintner, A., On the law of the iterated logarithm, American Journal of Mathematics, 1941, 63(1): 169−176. doi: 10.2307/2371287.

[7]

Hu, M. and Li, X., Independence under the G-expectation framework, J. Theor. Probab., 2014, 27: 1011−1020. doi: 10.1007/s10959-012-0471-y.

[8]

Kolmogorov, A., Über das Gesetz des iterierten Logarithmus, Mathematische Annalen, 1929, 101: 126–135.

[9]

Ledoux, M. and Talagrand, M., Probability in Banach Spaces: Isoperimetry and Processes, Springer Science & Business Media, 2013.

[10]

Li, S., Li, X. and Yuan, X., Upper and lower variances under model uncertainty and their applications in finance, International Journal of Financial Engineering, 2022, https://doi.org/10.1142/S2424786322500074.

[11]

Li, X. and Lin, Y., Generalized Wasserstein distance and weak convergence of sublinear expectations, J. Theor. Probab., 2017, 30: 581−593. doi: 10.1007/s10959-015-0651-7.

[12]

Li, X., On the functional central limit theorem with mean-uncertainty, arXiv: 2203.00170, 2022.

[13]

Peng, S., Law of large numbers and central limit theorem under nonlinear expectations, Probab. Uncertain. Quant. Risk, 2019, 4: 4, doi: 10.1186/s41546-019-0038-2.

[14]

Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty, Springer, Berlin, Heidelberg, 2019.

[15]

Stout, W. F., A martingale analogue of Kolmogorov’s law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie verw Gebiete, 1970, 15: 279–290.

[16]

Stout, W. F., The Hartman-Wintner law of the iterated logarithm for martingales, The Annals of Mathematical Statistics, 1970, 41(6): 2158−2160. doi: 10.1214/aoms/1177696721.

[17]

Stroock, D. W., Probability Theory: An Analytic View, Cambridge University Press, 1995.

[18]

Walley, P., Statistic Reasoning with Imprecise Probabilities, Chapman and Hall, 1993.

[19]

Zhang, L. X., Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm, Science China Mathematics, 2016, 59(12): 2503−2526. doi: 10.1007/s11425-016-0079-1.

[20]

Zhang, L. X., On the laws of the iterated logarithm under sub-linear expectations, Probab. Uncertain. Quant. Risk, 2021, 6(4): 409−460. doi: 10.3934/puqr.2021020.

show all references

References:
[1]

Chen, Z. and Hu, F., A law of the iterated logarithm under sublinear expectations, Journal of Financial Engineering, 2014, 1(2): 1450015. doi: 10.1142/S2345768614500159.

[2]

Feller, W., The general form of the so-called law of the iterated logarithm, Transactions of the American Mathematical Society, 1943, 54(3): 373−402. doi: 10.1090/S0002-9947-1943-0009263-7.

[3]

Hu, M., Li, X. and Li, X., Convergence rate of Peng’s law of large numbers under sublinear expectations, Probab. Uncertain. Quant. Risk, 2021, 6(3): 261−266. doi: 10.3934/puqr.2021013.

[4]

Guo, X., Li, S. and Li, X., On the Hartman-Wintner law of the iterated logarithm under sublinear expectation, Communications in Statistics-Theory and Methods, 2022, https://doi.org/10.1080/03610926.2022.2026394.

[5]

Guo, X. and Li, X., On the laws of large numbers for pseudo-independent random variables under sublinear expectation, Statistic and Probability Letters, 2021, 172: 109042. doi: 10.1016/j.spl.2021.109042.

[6]

Hartman, P. and Wintner, A., On the law of the iterated logarithm, American Journal of Mathematics, 1941, 63(1): 169−176. doi: 10.2307/2371287.

[7]

Hu, M. and Li, X., Independence under the G-expectation framework, J. Theor. Probab., 2014, 27: 1011−1020. doi: 10.1007/s10959-012-0471-y.

[8]

Kolmogorov, A., Über das Gesetz des iterierten Logarithmus, Mathematische Annalen, 1929, 101: 126–135.

[9]

Ledoux, M. and Talagrand, M., Probability in Banach Spaces: Isoperimetry and Processes, Springer Science & Business Media, 2013.

[10]

Li, S., Li, X. and Yuan, X., Upper and lower variances under model uncertainty and their applications in finance, International Journal of Financial Engineering, 2022, https://doi.org/10.1142/S2424786322500074.

[11]

Li, X. and Lin, Y., Generalized Wasserstein distance and weak convergence of sublinear expectations, J. Theor. Probab., 2017, 30: 581−593. doi: 10.1007/s10959-015-0651-7.

[12]

Li, X., On the functional central limit theorem with mean-uncertainty, arXiv: 2203.00170, 2022.

[13]

Peng, S., Law of large numbers and central limit theorem under nonlinear expectations, Probab. Uncertain. Quant. Risk, 2019, 4: 4, doi: 10.1186/s41546-019-0038-2.

[14]

Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty, Springer, Berlin, Heidelberg, 2019.

[15]

Stout, W. F., A martingale analogue of Kolmogorov’s law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie verw Gebiete, 1970, 15: 279–290.

[16]

Stout, W. F., The Hartman-Wintner law of the iterated logarithm for martingales, The Annals of Mathematical Statistics, 1970, 41(6): 2158−2160. doi: 10.1214/aoms/1177696721.

[17]

Stroock, D. W., Probability Theory: An Analytic View, Cambridge University Press, 1995.

[18]

Walley, P., Statistic Reasoning with Imprecise Probabilities, Chapman and Hall, 1993.

[19]

Zhang, L. X., Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm, Science China Mathematics, 2016, 59(12): 2503−2526. doi: 10.1007/s11425-016-0079-1.

[20]

Zhang, L. X., On the laws of the iterated logarithm under sub-linear expectations, Probab. Uncertain. Quant. Risk, 2021, 6(4): 409−460. doi: 10.3934/puqr.2021020.

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