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January  2017, 2: 4 doi: 10.1186/s41546-017-0013-8

Convergence to a self-normalized G-Brownian motion

Zhengyan Lin and  Li-Xin Zhang ,

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027 China

Received  October 25, 2016 Revised  January 05, 2017 Published  June 2017

Fund Project: supported by Grants from the National Natural Science Foundation of China (No. 11225104), the 973 Program (No. 2015CB352302) and the Fundamental Research Funds for the Central Universities.

G-Brownian motion has a very rich and interesting new structure that nontrivially generalizes the classical Brownian motion. Its quadratic variation process is also a continuous process with independent and stationary increments. We prove a self-normalized functional central limit theorem for independent and identically distributed random variables under the sub-linear expectation with the limit process being a G-Brownian motion self-normalized by its quadratic variation. To prove the self-normalized central limit theorem, we also establish a new Donsker's invariance principle with the limit process being a generalized G-Brownian motion.
Keywords: Sub-linear expectation, G-Brownian motion, Central limit theorem, Invariance principle, Self-normalization.
Mathematics Subject Classification: 60F15;60F05;60H10;60G48.
Citation: Zhengyan Lin, Li-Xin Zhang. Convergence to a self-normalized G-Brownian motion. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 4-. doi: 10.1186/s41546-017-0013-8
References:
[1]

Csörgö, M, Szyszkowicz, B, Wang, QY:Donsker's theorem for self-normalized partial sums processes. Ann. Probab 31, 1228-1240 (2003)

[2]

Denis, L, Hu, MS, Peng, SG:Function spaces and capacity related to a sublinear expectation:application to G-Brownian Motion Paths. Potential Anal 34, 139-161 (2011). arXiv:0802.1240v1[math.PR]

[3]

Giné, E, Götze, F, Mason, DM:When is the Student t-statistic asymptotically standard normal? Ann.Probab 25, 1514-1531 (1997)

[4]

Hu, MS, Ji, SL, Peng, SG, Song, YS:Backward stochastic differential equations driven by G-Brownian motion. Stochastic Process. Appl 124(1), 759-784 (2014a)

[5]

Hu, MS, Ji, SL, Peng, SG, Song, YS:Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion. Stochastic Process. Appl 124(2), 1170-1195(2014b)

[6]

Li, XP, Peng, SG:Topping times and related Ito's calculus with G-Brownian motion. Stochastic Process.Appl 121(7), 1492-1508 (2011)

[7]

Nutz, M, van Handel, R:Constructing sublinear expectations on path space. Stochastic Process. Appl 123(8), 3100-3121 (2013)

[8]

Peng, SG:G-expectation, G-Brownian motion and related stochastic calculus of Ito's type. The Abel Symposium 2005, Abel Symposia 2, Edit. Benth et. al, pp. 541-567. Springer-Verlag (2006)

[9]

Peng, SG:Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation.Stochastic Process. Appl 118(12), 2223-2253 (2008a)

[10]

Peng, SG:A new central limit theorem under sublinear expectations (2008b). Preprint:arXiv:0803.2656v1[math.PR]

[11]

Peng, SG:Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations. Sci. China Ser. A 52(7), 1391-1411 (2009)

[12]

Peng, SG:Nonlinear Expectations and Stochastic Calculus under Uncertainty (2010a). Preprint:arXiv:1002.4546[math.PR]

[13]

Peng, SG:Tightness, weak compactness of nonlinear expectations and application to CLT (2010b).Preprint:arXiv:1006.2541[math.PR]

[14]

Yan, D, Hutz, M, Soner, HM:Weak approximation of G-expectations. Stochastic Process. Appl 122(2), 664-675 (2012)

[15]

Zhang, LX:Donsker's invariance principle under the sub-linear expectation with an application to Chung's law of the iterated logarithm. Commun. Math. Stat 3(2), 187-214 (2015). arXiv:1503.02845[math.PR]

[16]

Zhang, LX:Rosenthal's inequalities for independent and negatively dependent random variables under sub-linear expectations with applications. Sci. China Math 59(4), 751-768 (2016)

show all references

References:
[1]

Csörgö, M, Szyszkowicz, B, Wang, QY:Donsker's theorem for self-normalized partial sums processes. Ann. Probab 31, 1228-1240 (2003)

[2]

Denis, L, Hu, MS, Peng, SG:Function spaces and capacity related to a sublinear expectation:application to G-Brownian Motion Paths. Potential Anal 34, 139-161 (2011). arXiv:0802.1240v1[math.PR]

[3]

Giné, E, Götze, F, Mason, DM:When is the Student t-statistic asymptotically standard normal? Ann.Probab 25, 1514-1531 (1997)

[4]

Hu, MS, Ji, SL, Peng, SG, Song, YS:Backward stochastic differential equations driven by G-Brownian motion. Stochastic Process. Appl 124(1), 759-784 (2014a)

[5]

Hu, MS, Ji, SL, Peng, SG, Song, YS:Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion. Stochastic Process. Appl 124(2), 1170-1195(2014b)

[6]

Li, XP, Peng, SG:Topping times and related Ito's calculus with G-Brownian motion. Stochastic Process.Appl 121(7), 1492-1508 (2011)

[7]

Nutz, M, van Handel, R:Constructing sublinear expectations on path space. Stochastic Process. Appl 123(8), 3100-3121 (2013)

[8]

Peng, SG:G-expectation, G-Brownian motion and related stochastic calculus of Ito's type. The Abel Symposium 2005, Abel Symposia 2, Edit. Benth et. al, pp. 541-567. Springer-Verlag (2006)

[9]

Peng, SG:Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation.Stochastic Process. Appl 118(12), 2223-2253 (2008a)

[10]

Peng, SG:A new central limit theorem under sublinear expectations (2008b). Preprint:arXiv:0803.2656v1[math.PR]

[11]

Peng, SG:Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations. Sci. China Ser. A 52(7), 1391-1411 (2009)

[12]

Peng, SG:Nonlinear Expectations and Stochastic Calculus under Uncertainty (2010a). Preprint:arXiv:1002.4546[math.PR]

[13]

Peng, SG:Tightness, weak compactness of nonlinear expectations and application to CLT (2010b).Preprint:arXiv:1006.2541[math.PR]

[14]

Yan, D, Hutz, M, Soner, HM:Weak approximation of G-expectations. Stochastic Process. Appl 122(2), 664-675 (2012)

[15]

Zhang, LX:Donsker's invariance principle under the sub-linear expectation with an application to Chung's law of the iterated logarithm. Commun. Math. Stat 3(2), 187-214 (2015). arXiv:1503.02845[math.PR]

[16]

Zhang, LX:Rosenthal's inequalities for independent and negatively dependent random variables under sub-linear expectations with applications. Sci. China Math 59(4), 751-768 (2016)

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