# American Institute of Mathematical Sciences

December  2021, 6(4): 409-460. doi: 10.3934/puqr.2021020

## On the laws of the iterated logarithm under sub-linear expectations

 1 School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, Zhejiang, China

Received  March 03, 2021 Accepted  November 11, 2021 Published  December 2021

Fund Project: Thanks to Professor Mingshang Hu for the constructive discussion which improved our original manuscript and the revision. Special thanks to the anonymous referees for carefully reading the manuscript and constructive comments. An example given by the referees led us to consider carefully the relationship between the capacity ${\widehat{\mathbb V}^{\ast}}$ and the probability measure, and the properties of ${\widehat{\mathbb V}^{\ast}}$. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11731012, 12031005), Ten Thousand Talents Plan of Zhejiang Province (Grant No. 2018R52042), Natural Science Foundation of Zhejiang Province (Grant No. LZ21A010002), and the Fundamental Research Funds for the Central Universities.

In this paper, we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space, where the random variables are not necessarily identically distributed. Exponential inequalities for the maximum sum of independent random variables and Kolmogorov’s converse exponential inequalities are established as tools for showing the law of the iterated logarithm. As an application, the sufficient and necessary conditions of the law of the iterated logarithm for independent and identically distributed random variables under the sub-linear expectation are obtained. In the paper, it is also shown that if the sub-linear expectation space is rich enough, it will have no continuous capacity. The laws of the iterated logarithm are established without the assumption on the continuity of capacities.

Citation: Li-Xin Zhang. On the laws of the iterated logarithm under sub-linear expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (4) : 409-460. doi: 10.3934/puqr.2021020
##### References:
 [1] Billingsley, P., Convergence of Probability Measures, 2nd ed., John Wiley &Sons, New York, 1999. [2] Chen, Z. J. and Hu, F., A law of the iterated logarithm under sublinear expectations, Journal of Financial Engineering, arXiv: 1103.2965v2, 2014, 1(2): 1450015. [3] Chen, X., On the law of the iterated logarithm for independent Banach space valued random variables, Ann. Probab., 1993, 21(4): 1991−2011. [4] Fuk, D. K. and Nagaev, S. V., Probability inequalities for sums of independent random variables, Theory of Probability & Its Applications, 1971, 16(4): 643−660. [5] Hartman, P. and Wintner, A., On the law of iterated logarithm, Amer. J. Math., 1941, 63(1): 169−176. doi: 10.2307/2371287. [6] Loève, M., Probability Theory I, 4th ed., Springer, New York, 1977. [7] Kolmogorov, A., Über das Gesetz des iterierten Logarithmus, Math. Ann., 1929, 101: 126–135. [8] Peng, S., A new central limit theorem under sublinear expectations, arXiv: 0803.2656v1, 2008. [9] Peng S., Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations, Sci. China Ser. A, 2009, 52(7): 1391−1411. [10] Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty with Robust CLT and G-Brownian Motion, Springer, Berlin, Heidelberg, 2019. [11] Peng, S., Yang, S. Z. and Yao, J. F., Improving Value-at-Risk prediction under model uncertainty, arXiv: 1805.03890, 2020. [12] Peng, S. and Zhou, Q., A hypothesis-testing perspective on the G-normal distribution theory, Stat. & Probab. Lett., 2020, 156: 108623, https://doi.org/10.1016/j.spl.2019.108623. [13] Petrov, V. V., Limit Theorem of Probability Theory-Sequences of independent Random Variables, Clarendon Press, Oxford, 1995. [14] Stout, W. F., Almost Sure Convergence, Academic Press, New York, 1974. [15] Strassen, V., A converse of the law of the iterated logarithm, Z. Wahrsch. Verw. Geb., 1966, 4: 265–268. [16] Wittmann, R., A general law of iterated logarithm, Z. Wahrsch. Verw. Geb, 1985, 68: 521–543. [17] Wittmann, R., Sufficient moment and truncated moment conditions for the law of iterated logarithm, Probab. Th. Rel. Fields, 1987, 75: 509−530. doi: 10.1007/BF00320331. [18] Yan, S. J., Wang, J. X. and Liu, X. F., Fundamentals of Probability Theory (in Chinese), Science press, Beijing, 1997. [19] 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. doi: 10.1007/s11425-016-0079-1. [20] Zhang, L. X., The convergence of the sums of independent random variables under the sub-linear expectations, Acta Mathematica Sinica, English Series, 2020, 36(3): 224−244. doi: 10.1007/s10114-020-8508-0. [21] Zhang, L. X., Lindeberg's central limit theorems for martingale like sequences under sublinear expectations, Sci. China Math., 2021, 64(6): 1263−1290. doi: 10.1007/s11425-018-9556-7.

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##### References:
 [1] Billingsley, P., Convergence of Probability Measures, 2nd ed., John Wiley &Sons, New York, 1999. [2] Chen, Z. J. and Hu, F., A law of the iterated logarithm under sublinear expectations, Journal of Financial Engineering, arXiv: 1103.2965v2, 2014, 1(2): 1450015. [3] Chen, X., On the law of the iterated logarithm for independent Banach space valued random variables, Ann. Probab., 1993, 21(4): 1991−2011. [4] Fuk, D. K. and Nagaev, S. V., Probability inequalities for sums of independent random variables, Theory of Probability & Its Applications, 1971, 16(4): 643−660. [5] Hartman, P. and Wintner, A., On the law of iterated logarithm, Amer. J. Math., 1941, 63(1): 169−176. doi: 10.2307/2371287. [6] Loève, M., Probability Theory I, 4th ed., Springer, New York, 1977. [7] Kolmogorov, A., Über das Gesetz des iterierten Logarithmus, Math. Ann., 1929, 101: 126–135. [8] Peng, S., A new central limit theorem under sublinear expectations, arXiv: 0803.2656v1, 2008. [9] Peng S., Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations, Sci. China Ser. A, 2009, 52(7): 1391−1411. [10] Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty with Robust CLT and G-Brownian Motion, Springer, Berlin, Heidelberg, 2019. [11] Peng, S., Yang, S. Z. and Yao, J. F., Improving Value-at-Risk prediction under model uncertainty, arXiv: 1805.03890, 2020. [12] Peng, S. and Zhou, Q., A hypothesis-testing perspective on the G-normal distribution theory, Stat. & Probab. Lett., 2020, 156: 108623, https://doi.org/10.1016/j.spl.2019.108623. [13] Petrov, V. V., Limit Theorem of Probability Theory-Sequences of independent Random Variables, Clarendon Press, Oxford, 1995. [14] Stout, W. F., Almost Sure Convergence, Academic Press, New York, 1974. [15] Strassen, V., A converse of the law of the iterated logarithm, Z. Wahrsch. Verw. Geb., 1966, 4: 265–268. [16] Wittmann, R., A general law of iterated logarithm, Z. Wahrsch. Verw. Geb, 1985, 68: 521–543. [17] Wittmann, R., Sufficient moment and truncated moment conditions for the law of iterated logarithm, Probab. Th. Rel. Fields, 1987, 75: 509−530. doi: 10.1007/BF00320331. [18] Yan, S. J., Wang, J. X. and Liu, X. F., Fundamentals of Probability Theory (in Chinese), Science press, Beijing, 1997. [19] 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. doi: 10.1007/s11425-016-0079-1. [20] Zhang, L. X., The convergence of the sums of independent random variables under the sub-linear expectations, Acta Mathematica Sinica, English Series, 2020, 36(3): 224−244. doi: 10.1007/s10114-020-8508-0. [21] Zhang, L. X., Lindeberg's central limit theorems for martingale like sequences under sublinear expectations, Sci. China Math., 2021, 64(6): 1263−1290. doi: 10.1007/s11425-018-9556-7.
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