非线性概率下行内负相协随机变量阵列的对数律
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摘要
本文在上概率空间中给出随机变量负相协的定义,该定义弱于现有非线性概率下的某些独立性概念.在此框架下,本文通过对随机变量阵列收敛性质的研究,得到上概率下行内负相协随机变量阵列的对数律,并同时给出依容度收敛的弱对数律.
In this paper, the notion of negatively associated random variables is introduced under nonlinear probabilities, which is weaker than some existing conditions of independence. Within this framework, a law of the logarithm for arrays of rowwise negatively associated random variables is established by investigating their convergence properties. In addition, a weak law of the logarithm with the convergence of capacity is obtained.
In this paper, the notion of negatively associated random variables is introduced under nonlinear probabilities, which is weaker than some existing conditions of independence. Within this framework, a law of the logarithm for arrays of rowwise negatively associated random variables is established by investigating their convergence properties. In addition, a weak law of the logarithm with the convergence of capacity is obtained.
引文
1 Chen Z,Epstein L.Ambiguity,risk,and asset returns in continuous time.Econometrica,2002,70:1403-1443
2 Peng S.G-expectation,G-Brownian motion and related stochastic calculus of Ito type.In:Stochastic Analysis and Applications.New York:Springer,2007,541-567
3 Peng S.Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and G-Brownian motion.ArXiv:1002.4546,2010
4 Choquet G.Theory of capacities.Ann Inst Fourier(Grenoble),1954,5:131-295
5 Chen Z,Chen T,Davison M.Choquet expectation and Peng’s g-expectation.Ann Probab,2005,33:1179-1199
6 Huper P.The use of Choquet capacities in statistics.Bull Internat Statist Inst,1973,45:181-191
7 Wasserman L,Kadane J.Bayes’theorem for Choquet capacities.Ann Statist,1990,18:1328-1339
8 Maccheroni F,Marinacci M.A strong law of large numbers for capacities.Ann Probab,2005,33:1171-1178
9 Marinacci M.Limit laws for non-additive probabilities and their frequentist interpretation.J Econom Theory,1999,84:145-195
10 Epstein L,Schneider M.IID:Independently and indistinguishably distributed.J Econom Theory,2003,113:32-50
11 De Cooman G,Miranda E.Weak and strong laws of large numbers for coherent lower previsions.J Statist Plann Inference,2008,138:2409-2432
12 Cozman F.Concentration inequalities and laws of large numbers under epistemic and regular irrelevance.Internat JApprox Reason,2010,51:1069-1084
13 Teran P.Laws of large numbers without additivity.Trans Amer Math Soc,2014,366:5431-5451
14 Zhang L.Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications.Sci China Math,2016,59:751-768
15 Chen Z,Hu F.A law of the iterated logarithm under sublinear expectations.J Financ Eng,2014,1:1450015
16 Zhang L.Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm.Sci China Math,2016,59:2503-2526
17 Zhang L.Strong limit theorems for extend independent and extended negatively dependent random variables under non-linear expecations.ArXiv:1680.00710,2016
18 Chen Z,Wu P,Li B.A strong law of large numbers for non-additive probabilities.Internat J Approx Reason,2013,54:365-377
19 Hu T,Moricz F,Taylor R.Strong laws of large numbers for arrays of rowwise independent random variables.Acta Math Hungar,1989,54:153-162
20 Hu T,Weber N.On the rate of convergence in the strong law of large numbers for arrays.Bull Aust Math Soc,1992,45:479-482
21 Lan Y,Zhang N.Strong limit theorems for weighted sums of negatively associated random variables in nonlinear probability.ArXiv:1706.05788,2017
22 Block H,Savits T,Shaked M.Some concepts of negative dependence.Ann Probab,1982,10:765-772
1)Feng X,Lan Y.Strong limit theorems for arrays of rowwise independent random variables under sublinear expectations.Submitted
2 Peng S.G-expectation,G-Brownian motion and related stochastic calculus of Ito type.In:Stochastic Analysis and Applications.New York:Springer,2007,541-567
3 Peng S.Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and G-Brownian motion.ArXiv:1002.4546,2010
4 Choquet G.Theory of capacities.Ann Inst Fourier(Grenoble),1954,5:131-295
5 Chen Z,Chen T,Davison M.Choquet expectation and Peng’s g-expectation.Ann Probab,2005,33:1179-1199
6 Huper P.The use of Choquet capacities in statistics.Bull Internat Statist Inst,1973,45:181-191
7 Wasserman L,Kadane J.Bayes’theorem for Choquet capacities.Ann Statist,1990,18:1328-1339
8 Maccheroni F,Marinacci M.A strong law of large numbers for capacities.Ann Probab,2005,33:1171-1178
9 Marinacci M.Limit laws for non-additive probabilities and their frequentist interpretation.J Econom Theory,1999,84:145-195
10 Epstein L,Schneider M.IID:Independently and indistinguishably distributed.J Econom Theory,2003,113:32-50
11 De Cooman G,Miranda E.Weak and strong laws of large numbers for coherent lower previsions.J Statist Plann Inference,2008,138:2409-2432
12 Cozman F.Concentration inequalities and laws of large numbers under epistemic and regular irrelevance.Internat JApprox Reason,2010,51:1069-1084
13 Teran P.Laws of large numbers without additivity.Trans Amer Math Soc,2014,366:5431-5451
14 Zhang L.Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications.Sci China Math,2016,59:751-768
15 Chen Z,Hu F.A law of the iterated logarithm under sublinear expectations.J Financ Eng,2014,1:1450015
16 Zhang L.Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm.Sci China Math,2016,59:2503-2526
17 Zhang L.Strong limit theorems for extend independent and extended negatively dependent random variables under non-linear expecations.ArXiv:1680.00710,2016
18 Chen Z,Wu P,Li B.A strong law of large numbers for non-additive probabilities.Internat J Approx Reason,2013,54:365-377
19 Hu T,Moricz F,Taylor R.Strong laws of large numbers for arrays of rowwise independent random variables.Acta Math Hungar,1989,54:153-162
20 Hu T,Weber N.On the rate of convergence in the strong law of large numbers for arrays.Bull Aust Math Soc,1992,45:479-482
21 Lan Y,Zhang N.Strong limit theorems for weighted sums of negatively associated random variables in nonlinear probability.ArXiv:1706.05788,2017
22 Block H,Savits T,Shaked M.Some concepts of negative dependence.Ann Probab,1982,10:765-772
1)Feng X,Lan Y.Strong limit theorems for arrays of rowwise independent random variables under sublinear expectations.Submitted