针对不确定变量分布函数的问题，提出了两个不确定中心极限定理.定义了不确定变量的特征函数并基于期望计算法则提出了特征函数的计算方法.分析了不确定变量特征函数的性质.将随机理论中的正态分布引入到不确定理论中，证实了该分布在形式上为一种正则不确定分布.通过新型坦克射击测试的案例验证了所提定理的可行性和有效性.
In order to determine the distribution of uncertain variables, two central limit theorems for independent and identical variables are proposed. Firstly, the characteristic function of uncertain variables is defined and its calculation formula is given based on the expected value of uncertain variables. Secondly, the properties of the characteristic function are analyzed and the normal distribution in probability theory is introduced into uncertainty theory. It is proved that the distribution can also be regarded as an uncertain one. Finally, an example of shooting accuracy of a new kind of tanks moving at a high speed is given to verify the feasibility and effectiveness of the proposed theorems.
［1］PENG Z, IWAMURA K. A sufficient and necessary condition of uncertainty distribution ［J］. Journal of Interdisciplinary Mathematics, 2010, 13(3): 277-285.
［2］WANG X, GAO Z, GUO H. Delphi method for estimating uncertainty distributions ［J］. International Journal on Information, 2012, 15(2): 449-460.
［3］DOLGOPYAT D, GOLDSHEID I. Central limit theorem for recurrent random walks on a strip with bounded potential ［J］. Nonlinearity, 2018, 31(7): 3381-3412.
［4］BENOIST Y, QUINT J F. Central limit theorem for linear groups［J］. The Annals of Probability, 2016, 44(2): 1308-1340.
［5］O’ROURKE S, RENFREW D. Central limit theorem for linear eigenvalue statistics of elliptic random matrices［J］. Journal of Theoretical Probability, 2016, 29(3): 1121-1191.
［6］GUO J, WANG Z, ZHENG M, et al. Uncertain multiobjective redundancy allocation problem of repairable systems based on artificial bee colony algorithm ［J］. Chinese Journal of Aeronautics, 2014, 27(6): 1477-1487.
［7］LIU L, ZHANG B, MA W. Uncertain programming models for fixed charge multi-item solid transportation problem ［J］. Soft Computing, 2018, 22(17): 5825-5833.
［8］DALMAN H. Uncertain programming model for multi-item solid transportation problem ［J］. International Journal of Machine Learning and Cybernetics, 2018, 9(4): 559-567.
［9］CHEN X. Uncertain calculus with finite variation processes ［J］. Soft Computing, 2015, 19(10): 2905-2912.
［10］LIU B, CHEN X. Uncertain multiobjective programming and uncertain goal programming ［J］. Journal of Uncertainty Analysis and Applications, 2015, 3(1): 1-8.
［11］LIU B, YAO K. Uncertain multilevel programming: Algorithm and applications ［J］. Computers & Industrial Engineering, 2015, 89: 235-240.
［12］PENG J. Risk metrics of loss function for uncertain system ［J］. Fuzzy Optimization and Decision Making, 2013, 12(1): 53-64.
［13］ZHANG Z, LIU W. Geometric average Asian option pricing for uncertain financial market ［J］. Journal of Uncertain Systems, 2014, 8(4): 317-320.
［14］WANG X, PENG Z. Method of moments for estimating uncertainty distributions ［J］. Journal of Uncertainty Analysis and Applications, 2014, 2 (1): 1-10.
［15］NING Y, CHEN X, WANG Z, et al. An uncertain multi-objective programming model for machine scheduling problem ［J］. International Journal of Machine Learning and Cybernetics, 2016, 8(5): 1493-1500.
［16］LI R, LIU G. An uncertain goal programming model for machine scheduling problem ［J］. Journal of Intelligent Manufacturing, 2017, 28(3): 689-694.
［17］王族统. 基于不确定理论的多目标规划方法及其应用研究［D］.西安: 空军工程大学,2015.
WANG Zutong. Research on multi-objective programming and apply based on uncertainty theory ［D］. Xi’an: Air Force Engineering University, 2015.
［18］LIU B. Uncertain random graph and uncertain random network ［J］. Journal of Uncertain Systems, 2014, 8(1): 3-12.
