可分B-空间中一般加权和的大数定律
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摘要
利用可分Banach空间中已有的概率不等式及对称化方法,研究了可分B-空间中加权系数具有某些较弱性质的加权和收敛问题,得到了B-值独立同分布随机元序列的这类加权和的强、弱大数定律成立的充分条件,对可分Banach空间中的Cesaro大数定律和欧拉大数定律进行了推广.同时,得到了实值独立同分布随机变量序列的这类加权和的强、弱大数定律成立的充分条件.
By the probability inequality and the symmetrization method obtained in a separable Banach space,convergence for the weighted sums whose weighting coefficients have some weaker natures in a separable Banach space is investigated,and the sufficient condition is obtained for the strong and weak law of large numbers of the weighted sums of a sequence of B-valued random elements that are independent and identically distributed,and the Cesaro law of large numbers and the Euler law of large numbers in a separable Banach space are spread.At the same time,the sufficient condition is obtained for the strong and weak law of large numbers of the weighted sums of a sequence of real-valued random variables that are independent and identically distributed.
By the probability inequality and the symmetrization method obtained in a separable Banach space,convergence for the weighted sums whose weighting coefficients have some weaker natures in a separable Banach space is investigated,and the sufficient condition is obtained for the strong and weak law of large numbers of the weighted sums of a sequence of B-valued random elements that are independent and identically distributed,and the Cesaro law of large numbers and the Euler law of large numbers in a separable Banach space are spread.At the same time,the sufficient condition is obtained for the strong and weak law of large numbers of the weighted sums of a sequence of real-valued random variables that are independent and identically distributed.
引文
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[2]SUNG S H.On the strong law of large numbers for pairwise i.i.d.random variables with general moment conditions[J].Statistics and Probability Letters,2013,83:1963-1968.DOI:10.1016/g.spl.2013.05.009.
[3]HUAN N V,QUANG N V.The doob inequality and strong law of large numbers for multidimensional arrays in general Banach spaces[J].Kybernetika Volume,2012,48:254-267.
[4]HEINKEL B.An infinite-dimensional law of large numbers in Cesaro’s sense[J].J Theoret Prob,1990,3(4):533-546.
[5]BOZORGNIZ A,RAO M B.On summability methods and limit theorems for Banach space valued random variables[J].Bull Inst Math Acad Sinica,1979,7(1):1-6.
[6]刘吉定.Banach空间中的Euler弱大数定律[J].武汉化工学院学报,1998,20(3):83-85.LIU J D.The Euler’s weak law of large numbers in Banach space[J].J Wuhan Inst Chem Tech,1998,20(3):83-85(Ch).
[7]刘吉定.实值鞅差序列配重和的大数定律[J].武汉化工学院学报,2000,22(2):75-76.LIU J D.Law of large numbers of weighted sums of real-valued martingale differences[J].J Wuhan Inst Chem Tech,2000,22(2):75-76(Ch).
[8]刘吉定,江世宏.经验过程的欧拉弱大数定律[J].华中理工大学学报,1998,26(12):89-90.LIU J D,JIANG S H.The Euler’s weak law of large numbers for empirical processes[J].J Huazhong Univ of Sci&Tech,1998,26(12):89-90(Ch).
[9]HOFFMANN-JORGENSEN J.Sums of independent banach space valued random variables[J].Studia Math,1974,52:159-186.
[10]de ACOSTA A.Strong exponential integrability of sums of independent B-valued random vectors[J].Prob Math Stat,1980,1:133-150.
[11]刘吉定,胡宗材.经验过程的Cesaro大数定律[J].吉林大学自然科学学报,1998(4):l-7.LIU J D,HU Z C.The Cesaro’s law of large numbers for empirical processes[J].Acta Scientiarum Naturalium Universitatis Jilinensis,1998(4):1-7(Ch).