Abstract
We obtain a general invariance principle of G-Brownian motion for the law of the iterated logarithm (LIL for short). For continuous bounded independent and identically distributed random variables in G-expectation space, we also give an invariance principle for LIL. In some sense, this result is an extension of the classical Strassen’s invariance principle to the case where probability measure is no longer additive. Furthermore, we give some examples as applications.
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Chen Z J, Hu F. A law of the iterated logarithm under sublinear expectations. J Finan Eng, 2014, 1: 1450015
Chen Z J, Wu P Y. Strong laws of large numbers for Bernoulli experiments under ambiguity. In: Nonlinear Mathematics for Uncertainty and its Applicatons. Advances in Intelligent and Soft Computing, vol. 100. Berlin-Heidelbeg: Springer, 2011, 19–30
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: 365–377
Denis L, Hu M S, Peng S G. Function spaces and capacity related to a sublinear expectation: Application to G-Brownian motion paths. Potential Anal, 2011, 34: 139–161
Einmahl U. Strong invariance principles for partial sums of independent random vectors. Ann Probab, 1987, 15: 1419–1440
Hartman P, Wintner A. On the law of the interated logarithm. Amer J Math, 1941, 63: 169–176
Hu F, Chen Z J, Zhang D F. How big are the increments of G-Brownian motion. Sci China Math, 2014, 57: 1687–1700
Khintchine A. Über einen Satz der Wahrscheinlichkeitsrechnung. Fund Math, 1924, 6: 9–20
Kolmogoroff A. Über das Gesetz des iterierten Logarithmus. Math Ann, 1929, 101: 126–135
McLeish D L. Invariance principles for dependent variables. Z Wahrscheinlichkeitstheorie Ver Gebiete, 1975, 32: 165–178
McLeish D L. On the invariance principle for nonstationary mixingales. Ann Probab, 1977, 5: 616–621
Peligrad M, Utev S. A new maximal inequality and invariance principle for stationary sequences. Ann Probab, 2005, 33: 798–815
Peng S G. G-expectation, G-Brownian motion and related stochastic calculus of Itô type. In: Benth F E, et al., eds. Stochastic Analysis and Applications. The Abel Symposium 2005. New York: Springer-Verlag, 2007, 541–567
Peng S G. G-Brownian motion and dynamic risk measure under volatility uncertainty. ArXiv:0711.2834, 2007
Peng S G. Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stochastic Process Appl, 2008, 118: 2223–2253
Peng S G. Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations. Sci China Ser A, 2009, 52: 1391–1411
Peng S G. Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and G-Brownian motion. ArXiv:1002.4546, 2010
Strassen V. An invariance principle for the law of the iterated logarithm. Z Wahrscheinlichkeitstheorie Ver Gebiete, 1964, 3: 211–226
Wu W B. Strong invariance principles for dependent random variables. Ann Probab, 2007, 35: 2294–2320
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Wu, P., Chen, Z. Invariance principles for the law of the iterated logarithm under G-framework. Sci. China Math. 58, 1251–1264 (2015). https://doi.org/10.1007/s11425-015-5002-8
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DOI: https://doi.org/10.1007/s11425-015-5002-8