若干随机序列的极限定理及条件弱鞅的不等式
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摘要
概率论是从数量上研究随机现象规律性的学科,理论严谨,应用广泛,发展迅速.概率极限理论是概率论的重要分支之一,是概率论其他分支和数理统计的非常重要的理论基础.强大数律和弱大数律是概率极限理论中的两个重要的研究内容.本文主要致力于研究几类随机变量序列的极限定理,如强大数律和弱大数律.
本文第二章研究了鞅差序列最大部分和的完全收敛性和完全矩收敛性,进而获得鞅差序列的Marcinkiewicz-Zygmund型强大数律的收敛速度.这些结果包含了Baum-Katz型定理和Hsu-Robbins型定理作为特殊情形,将Stocia(2007,2011)部分和的结果推广到最大部分和的情形并且扩展了参数的范围.此外,我们也讨论了鞅差序列随机加权和的完全矩收敛性,将非随机权推广到随机权的情形,同时获得了鞅差序列随机加权和的Marcinkiewicz-Zygmund型强大数律.
本文第三章讨论了AANA随机变量阵列加权和的完全收敛性,这些结果完善和改进了Baek et al.(2008)相应的结果.此外,我们在较广参数范围和较弱矩条件下获得了AANA随机变量阵列加权和的完全矩收敛性,作为应用,在较广参数范围和较弱矩条件下,AANA随机变量阵列的Baum-Katz型结果以及AANA随机变量序列的Marcinkiewicz-Zygmund型强大数律被获得.
本文第四章给出了一类一致可积的概念,并在此一致可积的条件下讨论了鞅差阵列,两两m-相依随机变量阵列和NOD随机变量阵列的矩收敛和弱大数律,推广和改进了Sung et al.(2008)的相应结果.作为应用,我们建立了误差满足此类一致可积条件下非参数回归模型回归函数估计量的矩相合性.
本文第五章给出了一类随机变量阵列的条件一致可积的概念,并在此一致可积条件下建立了几类条件相依随机变量阵列的条件矩收敛,这些结果推广和改进了Ordonez Cabrera&Volodin (2005)和Chandra&Goswami (2006)相应的结果,推广了Ordonez Cabrera et al.(2012)的结果.
本文最后一章建立了条件弱鞅和条件弱鞅函数的一些不等式,如最大值不等式、最小值不等式、Doob型不等式.利用条件期望的Fubini公式和条件弱鞅的不等式,得到了非负条件弱鞅的最大咖不等式以及基于凹的Young函数的条件弱鞅的最大值不等式.作为应用,条件PA随机变量序列的强大数律被获得.
Probability theory is a science of quantitatively studying regularity of ran-dom phenomena, which is theoretically rigorous, used widely, and developed rapidly. Probability limit theory is one of the important branches and an very essentially theoretical basis of other branches of probability and mathematical statistics. The strong laws of large numbers and the weak laws of large numbers are two important subjects studied of probability limit theory. This thesis fo-cuses mainly on limit theorems of some sequences of random variables, such as the strong laws of large numbers and the weak laws of large numbers.
In Chapter2, we investigate the complete convergence and complete mo-ment convergence of maximal partial sum for martingale differences, thus, we obtain the convergence rates in Marcinkiewicz-Zygmund-type strong law of large numbers for martingale differences. The results include Baum-Katz-type Theo-rem and Hsu-Robbins-type Theorem as special cases, generalize the results for the partial sum of Stocia (2007,2011) to the case of maximal partial sum and expand the scope of the parameters. In addition, we also discuss the complete moment convergence for randomly weighted sums of martingale differences, which generalize the non-random weights to the case of random weights. Meanwhile, Marcinkiewicz-Zygmund-type strong law of large numbers of randomly weighted sums for martingale differences is obtained.
In Chapter3, we study the complete convergence of weighted sums for array of rowwise AANA random variables, which complement and improve the corre-sponding ones of Baek et al.(2008). In addition, under the more extensive scope of parameters and the weaker moment conditions, the complete convergence and the complete moment convergence fof weighted sums for array of rowwise AANA random variables are obtained. As an application, Baum-Katz-type Theorem of array of rowwise AANA random variables and Marcinkiewicz-Zygmund-type strong law of large numbers of sequences of AANA random variables are pre-sented under the more extensive scope of parameters and the weaker moment conditions.
In Chapter4, the concept of a kind of uniform integrability is given and the moment convergence and the weak laws of large number for martingale dif-ferences, pairwise m-dependent sequences and NOD sequences are investigated under the conditions of this uniform integrability. The results extend and im-prove the corresponding ones of Sung et al.(2008). As an application, we obtain the moment consistency of estimators of regression functions under the nonpara-metric regression model under errors satisfying the conditions of this uniform integrability.
The concept of the conditional uniform integrability for array of random variables in Chapter5. We discuss the conditional moment convergence of some arrays of conditionally dependent random variables, which generalize and improve the corresponding ones of Ordonez Cabrera&Volodin (2005) and Chandra&Goswami (2006) and generalize the corresponding ones of Ordonez Cabrera et al.(2012).
In the last chapter of this thesis we investigate some inequalities of con-ditional demimartingales and the function of conditional demimartingales, such as maximal inequalities, minimal inequalities and Doob-type inequality. Using a formula of conditional expectation and a maximal inequality of conditional demi-martingales, the maximal Φ-inequalities of nonnegative conditional demimartin-gales and some maximal inequalities of concave Young functions for conditional demimartingales are obtained. As an application, the strong law of large numbers of sequences of conditional PA random variables is presented.
本文第二章研究了鞅差序列最大部分和的完全收敛性和完全矩收敛性,进而获得鞅差序列的Marcinkiewicz-Zygmund型强大数律的收敛速度.这些结果包含了Baum-Katz型定理和Hsu-Robbins型定理作为特殊情形,将Stocia(2007,2011)部分和的结果推广到最大部分和的情形并且扩展了参数的范围.此外,我们也讨论了鞅差序列随机加权和的完全矩收敛性,将非随机权推广到随机权的情形,同时获得了鞅差序列随机加权和的Marcinkiewicz-Zygmund型强大数律.
本文第三章讨论了AANA随机变量阵列加权和的完全收敛性,这些结果完善和改进了Baek et al.(2008)相应的结果.此外,我们在较广参数范围和较弱矩条件下获得了AANA随机变量阵列加权和的完全矩收敛性,作为应用,在较广参数范围和较弱矩条件下,AANA随机变量阵列的Baum-Katz型结果以及AANA随机变量序列的Marcinkiewicz-Zygmund型强大数律被获得.
本文第四章给出了一类一致可积的概念,并在此一致可积的条件下讨论了鞅差阵列,两两m-相依随机变量阵列和NOD随机变量阵列的矩收敛和弱大数律,推广和改进了Sung et al.(2008)的相应结果.作为应用,我们建立了误差满足此类一致可积条件下非参数回归模型回归函数估计量的矩相合性.
本文第五章给出了一类随机变量阵列的条件一致可积的概念,并在此一致可积条件下建立了几类条件相依随机变量阵列的条件矩收敛,这些结果推广和改进了Ordonez Cabrera&Volodin (2005)和Chandra&Goswami (2006)相应的结果,推广了Ordonez Cabrera et al.(2012)的结果.
本文最后一章建立了条件弱鞅和条件弱鞅函数的一些不等式,如最大值不等式、最小值不等式、Doob型不等式.利用条件期望的Fubini公式和条件弱鞅的不等式,得到了非负条件弱鞅的最大咖不等式以及基于凹的Young函数的条件弱鞅的最大值不等式.作为应用,条件PA随机变量序列的强大数律被获得.
Probability theory is a science of quantitatively studying regularity of ran-dom phenomena, which is theoretically rigorous, used widely, and developed rapidly. Probability limit theory is one of the important branches and an very essentially theoretical basis of other branches of probability and mathematical statistics. The strong laws of large numbers and the weak laws of large numbers are two important subjects studied of probability limit theory. This thesis fo-cuses mainly on limit theorems of some sequences of random variables, such as the strong laws of large numbers and the weak laws of large numbers.
In Chapter2, we investigate the complete convergence and complete mo-ment convergence of maximal partial sum for martingale differences, thus, we obtain the convergence rates in Marcinkiewicz-Zygmund-type strong law of large numbers for martingale differences. The results include Baum-Katz-type Theo-rem and Hsu-Robbins-type Theorem as special cases, generalize the results for the partial sum of Stocia (2007,2011) to the case of maximal partial sum and expand the scope of the parameters. In addition, we also discuss the complete moment convergence for randomly weighted sums of martingale differences, which generalize the non-random weights to the case of random weights. Meanwhile, Marcinkiewicz-Zygmund-type strong law of large numbers of randomly weighted sums for martingale differences is obtained.
In Chapter3, we study the complete convergence of weighted sums for array of rowwise AANA random variables, which complement and improve the corre-sponding ones of Baek et al.(2008). In addition, under the more extensive scope of parameters and the weaker moment conditions, the complete convergence and the complete moment convergence fof weighted sums for array of rowwise AANA random variables are obtained. As an application, Baum-Katz-type Theorem of array of rowwise AANA random variables and Marcinkiewicz-Zygmund-type strong law of large numbers of sequences of AANA random variables are pre-sented under the more extensive scope of parameters and the weaker moment conditions.
In Chapter4, the concept of a kind of uniform integrability is given and the moment convergence and the weak laws of large number for martingale dif-ferences, pairwise m-dependent sequences and NOD sequences are investigated under the conditions of this uniform integrability. The results extend and im-prove the corresponding ones of Sung et al.(2008). As an application, we obtain the moment consistency of estimators of regression functions under the nonpara-metric regression model under errors satisfying the conditions of this uniform integrability.
The concept of the conditional uniform integrability for array of random variables in Chapter5. We discuss the conditional moment convergence of some arrays of conditionally dependent random variables, which generalize and improve the corresponding ones of Ordonez Cabrera&Volodin (2005) and Chandra&Goswami (2006) and generalize the corresponding ones of Ordonez Cabrera et al.(2012).
In the last chapter of this thesis we investigate some inequalities of con-ditional demimartingales and the function of conditional demimartingales, such as maximal inequalities, minimal inequalities and Doob-type inequality. Using a formula of conditional expectation and a maximal inequality of conditional demi-martingales, the maximal Φ-inequalities of nonnegative conditional demimartin-gales and some maximal inequalities of concave Young functions for conditional demimartingales are obtained. As an application, the strong law of large numbers of sequences of conditional PA random variables is presented.
引文
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