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Weak laws of large numbers for sublinear expectation
1. | Department of Mathematics, Shandong University, Jinan 250100, China |
2. | School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Bank of Weifang, Jinan 250014, China |
In this paper we study the weak laws of large numbers for sublinear expectation. We prove that, without any moment condition, the weak laws of large numbers hold in the sense of convergence in capacity induced by some general sublinear expectations. For some specific sublinear expectation, for instance, mean deviation functional and one-side moment coherent risk measure, we also give weak laws of large numbers for corresponding capacity.
References:
[1] |
P. Artzner, F. Delbaen, J. M. Eber and D. Heath,
Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228.
doi: 10.1111/1467-9965.00068. |
[2] |
X. Chen and Z. Chen, Weak and strong limit theorems for stochastic processes under nonadditive probability, Abstract and Applied Analysis, 2014 (2014), Art. ID 645947, 7 pp.
doi: 10.1155/2014/645947. |
[3] |
X. Chen, Strong law of large numbers under an upper probability, Applied Mathematics, 3 (2012), 2056. |
[4] |
Z. Chen,
Strong laws of large numbers for sub-linear expectations, Science China Mathematics, 59 (2016), 945-954.
doi: 10.1007/s11425-015-5095-0. |
[5] |
Z. Chen, P. Wu and B. Li,
A strong law of large numbers for non-additive probabilities, International Journal of Approximate Reasoning, 54 (2013), 365-377.
doi: 10.1016/j.ijar.2012.06.002. |
[6] |
C. Gustave,
Theory of capacities, Annales de l'institut Fourier, 5 (1953), 131-295.
|
[7] |
G. De Cooman and E. Miranda,
Weak and strong laws of large numbers for coherent lower previsions, Journal of Statistical Planning and Inference, 138 (2008), 2409-2432.
doi: 10.1016/j.jspi.2007.10.020. |
[8] |
R. Durrett, Probability: Theory and Examples, Cambridge university press, 2010.
doi: 10.1017/CBO9780511779398. |
[9] |
H. Föllmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time, Walter de Gruyter, 2011. |
[10] |
P. Ghirardato,
On independence for non-additive measures with a Fubini theorem, J. Econom. Theory, 73 (1997), 261-291.
doi: 10.1006/jeth.1996.2241. |
[11] |
F. Maccheroni and M. Massimo,
A strong law of large numbers for capacities, Annals of Probability, 33 (2005), 1171-1178.
doi: 10.1214/009117904000001062. |
[12] |
S. Peng, Law of large numbers and central limit theorem under nonlinear expectations, Sci. China Ser. A, 52 (2009), 1391-1411, arXiv: math/0702358.
doi: 10.1007/s11425-009-0121-8. |
[13] |
S. Peng, Nonlinear expectations and stochastic calculus under uncertainty, preprint, arXiv: 1002.4546. |
[14] |
G. C. Pflug, W. Römisch, Modeling, Measuring and Managing Risk, Singapore: World Scientific, 2007.
doi: 10.1142/9789812708724. |
[15] |
L. S. Shapley,
A value for n-person games, Contributions to the Theory of Games, 2 (1953), 307-317.
|
[16] |
P. Terán,
Laws of large numbers without additivity, Transactions of American Mathematical Society, 366 (2014), 5431-5451.
doi: 10.1090/S0002-9947-2014-06053-4. |
show all references
References:
[1] |
P. Artzner, F. Delbaen, J. M. Eber and D. Heath,
Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228.
doi: 10.1111/1467-9965.00068. |
[2] |
X. Chen and Z. Chen, Weak and strong limit theorems for stochastic processes under nonadditive probability, Abstract and Applied Analysis, 2014 (2014), Art. ID 645947, 7 pp.
doi: 10.1155/2014/645947. |
[3] |
X. Chen, Strong law of large numbers under an upper probability, Applied Mathematics, 3 (2012), 2056. |
[4] |
Z. Chen,
Strong laws of large numbers for sub-linear expectations, Science China Mathematics, 59 (2016), 945-954.
doi: 10.1007/s11425-015-5095-0. |
[5] |
Z. Chen, P. Wu and B. Li,
A strong law of large numbers for non-additive probabilities, International Journal of Approximate Reasoning, 54 (2013), 365-377.
doi: 10.1016/j.ijar.2012.06.002. |
[6] |
C. Gustave,
Theory of capacities, Annales de l'institut Fourier, 5 (1953), 131-295.
|
[7] |
G. De Cooman and E. Miranda,
Weak and strong laws of large numbers for coherent lower previsions, Journal of Statistical Planning and Inference, 138 (2008), 2409-2432.
doi: 10.1016/j.jspi.2007.10.020. |
[8] |
R. Durrett, Probability: Theory and Examples, Cambridge university press, 2010.
doi: 10.1017/CBO9780511779398. |
[9] |
H. Föllmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time, Walter de Gruyter, 2011. |
[10] |
P. Ghirardato,
On independence for non-additive measures with a Fubini theorem, J. Econom. Theory, 73 (1997), 261-291.
doi: 10.1006/jeth.1996.2241. |
[11] |
F. Maccheroni and M. Massimo,
A strong law of large numbers for capacities, Annals of Probability, 33 (2005), 1171-1178.
doi: 10.1214/009117904000001062. |
[12] |
S. Peng, Law of large numbers and central limit theorem under nonlinear expectations, Sci. China Ser. A, 52 (2009), 1391-1411, arXiv: math/0702358.
doi: 10.1007/s11425-009-0121-8. |
[13] |
S. Peng, Nonlinear expectations and stochastic calculus under uncertainty, preprint, arXiv: 1002.4546. |
[14] |
G. C. Pflug, W. Römisch, Modeling, Measuring and Managing Risk, Singapore: World Scientific, 2007.
doi: 10.1142/9789812708724. |
[15] |
L. S. Shapley,
A value for n-person games, Contributions to the Theory of Games, 2 (1953), 307-317.
|
[16] |
P. Terán,
Laws of large numbers without additivity, Transactions of American Mathematical Society, 366 (2014), 5431-5451.
doi: 10.1090/S0002-9947-2014-06053-4. |
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