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. Google Scholar

[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. Google Scholar

[3]

Chen, X., On the law of the iterated logarithm for independent Banach space valued random variables, Ann. Probab., 1993, 21(4): 1991−2011. Google Scholar

[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. Google Scholar

[5]

Hartman, P. and Wintner, A., On the law of iterated logarithm, Amer. J. Math., 1941, 63(1): 169−176. doi: 10.2307/2371287.  Google Scholar

[6]

Loève, M., Probability Theory I, 4th ed., Springer, New York, 1977. Google Scholar

[7]

Kolmogorov, A., Über das Gesetz des iterierten Logarithmus, Math. Ann., 1929, 101: 126–135. Google Scholar

[8]

Peng, S., A new central limit theorem under sublinear expectations, arXiv: 0803.2656v1, 2008. Google Scholar

[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. Google Scholar

[10]

Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty with Robust CLT and G-Brownian Motion, Springer, Berlin, Heidelberg, 2019. Google Scholar

[11]

Peng, S., Yang, S. Z. and Yao, J. F., Improving Value-at-Risk prediction under model uncertainty, arXiv: 1805.03890, 2020. Google Scholar

[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. Google Scholar

[13]

Petrov, V. V., Limit Theorem of Probability Theory-Sequences of independent Random Variables, Clarendon Press, Oxford, 1995. Google Scholar

[14]

Stout, W. F., Almost Sure Convergence, Academic Press, New York, 1974. Google Scholar

[15]

Strassen, V., A converse of the law of the iterated logarithm, Z. Wahrsch. Verw. Geb., 1966, 4: 265–268. Google Scholar

[16]

Wittmann, R., A general law of iterated logarithm, Z. Wahrsch. Verw. Geb, 1985, 68: 521–543. Google Scholar

[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.  Google Scholar

[18]

Yan, S. J., Wang, J. X. and Liu, X. F., Fundamentals of Probability Theory (in Chinese), Science press, Beijing, 1997. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

Billingsley, P., Convergence of Probability Measures, 2nd ed., John Wiley &Sons, New York, 1999. Google Scholar

[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. Google Scholar

[3]

Chen, X., On the law of the iterated logarithm for independent Banach space valued random variables, Ann. Probab., 1993, 21(4): 1991−2011. Google Scholar

[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. Google Scholar

[5]

Hartman, P. and Wintner, A., On the law of iterated logarithm, Amer. J. Math., 1941, 63(1): 169−176. doi: 10.2307/2371287.  Google Scholar

[6]

Loève, M., Probability Theory I, 4th ed., Springer, New York, 1977. Google Scholar

[7]

Kolmogorov, A., Über das Gesetz des iterierten Logarithmus, Math. Ann., 1929, 101: 126–135. Google Scholar

[8]

Peng, S., A new central limit theorem under sublinear expectations, arXiv: 0803.2656v1, 2008. Google Scholar

[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. Google Scholar

[10]

Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty with Robust CLT and G-Brownian Motion, Springer, Berlin, Heidelberg, 2019. Google Scholar

[11]

Peng, S., Yang, S. Z. and Yao, J. F., Improving Value-at-Risk prediction under model uncertainty, arXiv: 1805.03890, 2020. Google Scholar

[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. Google Scholar

[13]

Petrov, V. V., Limit Theorem of Probability Theory-Sequences of independent Random Variables, Clarendon Press, Oxford, 1995. Google Scholar

[14]

Stout, W. F., Almost Sure Convergence, Academic Press, New York, 1974. Google Scholar

[15]

Strassen, V., A converse of the law of the iterated logarithm, Z. Wahrsch. Verw. Geb., 1966, 4: 265–268. Google Scholar

[16]

Wittmann, R., A general law of iterated logarithm, Z. Wahrsch. Verw. Geb, 1985, 68: 521–543. Google Scholar

[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.  Google Scholar

[18]

Yan, S. J., Wang, J. X. and Liu, X. F., Fundamentals of Probability Theory (in Chinese), Science press, Beijing, 1997. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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