次线性期望下的一般中心极限定理
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摘要
受Peng-中心极限定理的启发,本文主要应用G-正态分布的概念,放宽Peng-中心极限定理的条件,在次线性期望下得到形式更为一般的中心极限定理.首先,将均值条件E[X_n]=ε[X_n]=0放宽为|E[X_n]|+|ε[X_n]|=O(1/n);其次,应用随机变量截断的方法,放宽随机变量的2阶矩与2+δ阶矩条件;最后,将该定理的Peng-独立性条件进行放宽,得到卷积独立随机变量的中心极限定理.
Inspired by the central limit theorem established by Peng,we investigate the generalized central limit theorem under sublinear expectations based on three weaker conditions with the notion of G-normal distribution.Initially,the condition E[X_n]=ε[X_n]=0 is replaced by |E[X_n]|+|ε[X_n]|=O(1/n). Furthermore,the original2-nd and(2+δ)-th moments conditions are weakened through the truncation of random variables.Finally,we develop the theorem for convolutionary random variables,which can be seen as a generalization of Peng-independence.
Inspired by the central limit theorem established by Peng,we investigate the generalized central limit theorem under sublinear expectations based on three weaker conditions with the notion of G-normal distribution.Initially,the condition E[X_n]=ε[X_n]=0 is replaced by |E[X_n]|+|ε[X_n]|=O(1/n). Furthermore,the original2-nd and(2+δ)-th moments conditions are weakened through the truncation of random variables.Finally,we develop the theorem for convolutionary random variables,which can be seen as a generalization of Peng-independence.
引文
[1]Chareka P.,The central limit theorem for capacities,Statistics and Probability Letters,2009,79(12):1456-1462.
[2]Chen Z.,Chen T.,Davison M.,Choquet expectation and Peng's g-expectation,The Annals of Probability.,2005,33(3):1179-1199.
[3]Chen Z.Epstein L.,Ambiguity,risk and asset returns in continuous time,Econometrica,2002,70(4):1403-1443.
[4]Chen Z.,Hu F.,A law of the iterated logarithm under sublinear expectations,Journal of Financial Engineering, 2014,1:1450015.
[5]Choquet G.,Theory of capacities,Annales de I'institut Fourier,1954,5:131-295.
[6]Coquet F.,Hu Y.,Memin J.,et al.,Filtration-consistent nonlinear expectations and related g-expectations,Probability Theory and Related Fields,2002,123(1):1-27.
[7]Epstein L.,Seo K.,A central limit theorem for belief functions,2011,in preparation.
[8]Hu F.,Zhang D.,Central limit theorem for capacities,Comptes Rendus Mathematique,2010,348(19/20):1111-1114.
[9]Li M.,Shi Y.,A general central limit theorem under sublinear expectations,Science China Mathematics,2010,53(8):1989-1994.
[10]Peng S.,G-expectation,G-Brownian Motion and Related Stochastic Calculus of It Type,Stochastic Analysis and Applications,Springer,Berlin,2007:541-567.
[11]Peng S.,Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and G-Brownian motion,preprint,arXiv:1002.4546,2010.
[12]Schmeidler D.,Subjective probability and expected utility without additivity,Econometrica, 1989,57:571-587.
[13]Teran P.,Counterexamples to a central limit theorem and a weak law of large numbers for capacities,Statistics and Probability Letters,2015,96:185-189.
[14]Wakker P.,Testing and characterizing properties of nonadditive measures through violations of the sure-thing principle,Econometrica,2001,69(4):1039-1059.
[15]Wang L.,On the regularity theory of fully nonlinear parabolic equationsⅠ,Communications on Pure and Applied Mathematics,1992,45(1):27-76.
[16]Wang L.,On the regularity theory of fully nonlinear parabolic equations II,Communications on Pure and Applied Mathematics,1992,45(2):141-178.
[17]Wu P.,Chen Z.,Invariance principles for the law of the iterated logarithm under G-framework,Science in China-Mathematics,2015,58(6):1251-1264.
[18]Zhang D.,Chen Z.,A weighted central limit theorem under sublinear expectations,Communications in Statistics-Theory and Methods,2014,43(3):566-577.
[19]Zhang N.,Lan Y.,A comparison theorem under sublinear expectations and related limit theorems,arXiv:1710.01624math.PR,2017.
[20]Zong G.,Research about non-linear expectation and related questions(in Chinese),Dissertation for Doctoral Degree,2015:40-43.
[2]Chen Z.,Chen T.,Davison M.,Choquet expectation and Peng's g-expectation,The Annals of Probability.,2005,33(3):1179-1199.
[3]Chen Z.Epstein L.,Ambiguity,risk and asset returns in continuous time,Econometrica,2002,70(4):1403-1443.
[4]Chen Z.,Hu F.,A law of the iterated logarithm under sublinear expectations,Journal of Financial Engineering, 2014,1:1450015.
[5]Choquet G.,Theory of capacities,Annales de I'institut Fourier,1954,5:131-295.
[6]Coquet F.,Hu Y.,Memin J.,et al.,Filtration-consistent nonlinear expectations and related g-expectations,Probability Theory and Related Fields,2002,123(1):1-27.
[7]Epstein L.,Seo K.,A central limit theorem for belief functions,2011,in preparation.
[8]Hu F.,Zhang D.,Central limit theorem for capacities,Comptes Rendus Mathematique,2010,348(19/20):1111-1114.
[9]Li M.,Shi Y.,A general central limit theorem under sublinear expectations,Science China Mathematics,2010,53(8):1989-1994.
[10]Peng S.,G-expectation,G-Brownian Motion and Related Stochastic Calculus of It Type,Stochastic Analysis and Applications,Springer,Berlin,2007:541-567.
[11]Peng S.,Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and G-Brownian motion,preprint,arXiv:1002.4546,2010.
[12]Schmeidler D.,Subjective probability and expected utility without additivity,Econometrica, 1989,57:571-587.
[13]Teran P.,Counterexamples to a central limit theorem and a weak law of large numbers for capacities,Statistics and Probability Letters,2015,96:185-189.
[14]Wakker P.,Testing and characterizing properties of nonadditive measures through violations of the sure-thing principle,Econometrica,2001,69(4):1039-1059.
[15]Wang L.,On the regularity theory of fully nonlinear parabolic equationsⅠ,Communications on Pure and Applied Mathematics,1992,45(1):27-76.
[16]Wang L.,On the regularity theory of fully nonlinear parabolic equations II,Communications on Pure and Applied Mathematics,1992,45(2):141-178.
[17]Wu P.,Chen Z.,Invariance principles for the law of the iterated logarithm under G-framework,Science in China-Mathematics,2015,58(6):1251-1264.
[18]Zhang D.,Chen Z.,A weighted central limit theorem under sublinear expectations,Communications in Statistics-Theory and Methods,2014,43(3):566-577.
[19]Zhang N.,Lan Y.,A comparison theorem under sublinear expectations and related limit theorems,arXiv:1710.01624math.PR,2017.
[20]Zong G.,Research about non-linear expectation and related questions(in Chinese),Dissertation for Doctoral Degree,2015:40-43.