弱鞅和三类相依序列的概率不等式及极限定理
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摘要
相依序列极限理论在应用概率、统计、保险与金融数学、复杂性系统、可靠性理论、生存分析等领域都有着广泛的应用.本文主要致力于研究弱鞅和三类相依序列的概率不等式,如Bernstein型不等式,Hajek-Renyi型不等式,Chow型极大值不等式,Doob型极大值不等式,等等,利用这些概率不等式,研究若干相依序列的完全收敛性、几乎处处收敛性以及强收敛速度等方面的极限性质.
本文的第二章研究了(?)混合序列的矩不等式,利用此不等式,我们得到了(?)混合序列的Kolmogorov型收敛定理、三级数定理、强大数定律和强收敛速度,同时还给出了(?)混合序列概率不等式的一些新结果,例如Hajek-Renyi型不等式等,由此证明了相应的上确界的可积性.利用(?)混合序列的Bernstein型不等式,研究了(?)混合序列的完全收敛性和逆矩,其逆矩的渐近逼近推广并改进了文献Kaluszka和Okolewski中的定理3、胡舒合等中的定理2.1和定理2.3以及Wu等中的定理1(条件sup1≤i≤nEZi／Bn≤C1可以去掉).另外,我们指出文献Sun和Ling中关于Bahadur表示的一个证明错误,同时还给出了(?)混合样本分位数的Bahadur表示,我们获得的界要优于Ling中的界.
第三章研究了有界和无界NOD序列的指数型不等式及强收敛性质. NOD序列是一类包含独立序列和NA序列作为其特例的较为广泛的相依序列,我们在适当的矩条件下,建立起无界NOD序列的指数型不等式如下：其中是一个正实数序列,且满足由此结果,我们可进一步研究NOD序列的强收敛性质.我们的结果推广并改进了文献Kim和Kim,Nooghabi和Azarnoosh以及Xing等中有关NA序列的相应结果.
LNQD序列的概率不等式和强收敛速度是我们第四章研究的重点内容LNQD序列是也是一类包含独立序列和NA序列作为其特例的较为广泛的相依序列,但不同于NOD序列.利用LNQD序列的基本性质,我们给出了若干指数型不等式,如Bernstein型不等式等,由此可进一步研究其完全收敛性和几乎处处收敛性.利用LNQD序列的矩不等式,我们得到了LNQD序列的大偏差定理和Hajek-Renyi型不等式,并由此给出LNQD序列的强收敛速度和上确界的可积性.这些都是LNQD序列的新结果.
本文在最后一章主要研究弱(下)鞅及其凸函数的极值不等式和极限定理.弱鞅概念是在20世纪八十年代被提出来的,包含鞅作为其特例,并且均值为零的独立序列、PA序列和强正相依序列的部分和序列也是弱鞅.在本章我们做出了如下五方面的贡献：
·研究了弱(下)鞅及其凸函数的Chow型极大值不等式和Doob型极大值不等式,并且获得了一些概率不等式的新结果；
·建立了弱(下)鞅及其凸函数在0＜p＜1,p=1以及p＞1场合下的Doob型不等式,推广了文献Christofides和Wang中的相应结果；
·指出文献Harremoes中定理4的一个证明错误,并给出完整证明,同时我们还给出了弱鞅极小值不等式的一个新结果,由此证明了其中{Sn,n≥1}为一非负弱鞅,S1=1,γ(x)=x-1-log x,x>0上述结果推广了文献Harremoes中有关鞅的相应结果；
·给出了弱(下)鞅及其凸函数的一些新的强大数定律和强收敛速度,推广并改进了文献Christofides,Chow和Prakasa Rao中的相应结果；
·给出弱下鞅一致可积的一个等价条件.
The limit theory of dependent suquences has wide applications in applied probability, statistics, insurance and financial mathematics, complex systems, reliability theory, sur-vival analysis and other fields. This thesis focuses mainly on probability inequalities for demimartingale and three dependent sequences, such as Bernstein-type inequality, Hajek-Renyi-type inequality, Chow-type maximal inequality, Doob-type maximal inequality, and so forth. The limit theorems for demimartingale and three dependent sequences, such as complete convergence, almost sure convergence, strong growth rate, and so on, are also considered.
In Chapter 2, we study the moment inequality forφmixing suquence. By the moment inequality, we obtain the Kolmogorov-type convergence theorem, three series theorem, strong law of large numbers and strong growth rate forφmixing suquence. We also give some new results of probability inequalities forφmixing suquence, such as Hajek-Renyi-type inequality, and so on. As a consequence, the integrability of supremum forφmixing suquence can be proved. By the Bernstein-type inequality, we investigate the complete convergence and asymptotic approximation of inverse moment forφmixing suquence. The asymptotic approximation of inverse moment that we obtain generalizes and improves the results of Theorem 3 in Kaluszka and Okolewski, Theorems 2.1 and 2.3 in Hu et al. and Theorem 1 in Wu et al.(the condition sup1≤i≤n EZi/Bn≤C1 can be removed). In addition, we point out that there ia a mistake in the proof of the main results in Sun and Ling, and give the Bahadur representation for sample quantiles underφmixing sequence. We get better bound than that in Ling.
In Chapter 3, we investigate the exponential inequalities and strong convergence properties for bounded and unbounded NOD sequence. NOD sequence is a very broad class of dependent sequences, which contains independent sequence and NA sequence as special cases. The main purpose of this chapter is to establish the following exponential inequality for unbounded NOD sequence under suitable moment conditions: where is a sequence of positive numbers satisfying Furthermore, we study the strong convergence properties for NOD sequence, which gener-alize and improve the corresponding results for NA sequence in Kim and Kim, Nooghabi and Azarnoosh and Xing et al.
The probability inequalities and strong growth rate for LNQD sequence is the main target of Chapter 4. LNQD sequence is also a very broad class of dependent sequences, which contains independent sequence and NA sequence as special cases, but different from NOD sequence. By the Basic properties of LNQD sequence, we get some exponential inequalities, such as Bernstein-type inequality, and so forth, which can be applied to study the complete convergence and almost sure convergence. We also obtain the large deviations and Hajek-Renyi-type inequality from the moment inequality of LNQD suequence, which can be applied to prove the strong growth rate and the integrability of supremum. The results listed above are new results for LNQD suquence.
In the lase chapter of this thesis, we study the maximal inequalities and limit the-orem for demimartingale and its convex function. The concept of demimartingale was introduced in 1980s, which contains martingale as a special case. The partial sums of sequences of mean zero independent random variables, mean zero associated random vari-ables and mean zero strongly positive dependent random variables are all demimartingale. We make the following five contributions in this chapter:
●We study the Chow-type maximal inequality and Doob-type maximal inequality for demi(sub)martingale and its convex function. Some new results of probability inequalities are also obtained;
●We establish the Doob-type inequality for demi(sub)martingale and its convex function under the case of 01, which generalizes the corresponding results of Christofides and Wang;
●We point out that there is a mistake in the proof Theorem 4 in Harremoes, and give a complete proof. A new minimal inequality for demimartinge is also obtained, which can be applied to prove the folowing inequality: where {Sn,n≥1} is a nonnegative demimartingale, S1=1,γ(x)=x-1-logx,x>0. The result above generalizes the corresponding one of Harremoes;
●We give some new kinds of strong law of large numbers and strong growth rate for demi(sub)martingale and its convex function, which generalize and improve the corre-sponding ones of Christofides, Chow and Prakasa Rao;
●An equivalent condition of uniform integrability for demisubmartingales is pre-sented.
本文的第二章研究了(?)混合序列的矩不等式,利用此不等式,我们得到了(?)混合序列的Kolmogorov型收敛定理、三级数定理、强大数定律和强收敛速度,同时还给出了(?)混合序列概率不等式的一些新结果,例如Hajek-Renyi型不等式等,由此证明了相应的上确界的可积性.利用(?)混合序列的Bernstein型不等式,研究了(?)混合序列的完全收敛性和逆矩,其逆矩的渐近逼近推广并改进了文献Kaluszka和Okolewski中的定理3、胡舒合等中的定理2.1和定理2.3以及Wu等中的定理1(条件sup1≤i≤nEZi／Bn≤C1可以去掉).另外,我们指出文献Sun和Ling中关于Bahadur表示的一个证明错误,同时还给出了(?)混合样本分位数的Bahadur表示,我们获得的界要优于Ling中的界.
第三章研究了有界和无界NOD序列的指数型不等式及强收敛性质. NOD序列是一类包含独立序列和NA序列作为其特例的较为广泛的相依序列,我们在适当的矩条件下,建立起无界NOD序列的指数型不等式如下：其中是一个正实数序列,且满足由此结果,我们可进一步研究NOD序列的强收敛性质.我们的结果推广并改进了文献Kim和Kim,Nooghabi和Azarnoosh以及Xing等中有关NA序列的相应结果.
LNQD序列的概率不等式和强收敛速度是我们第四章研究的重点内容LNQD序列是也是一类包含独立序列和NA序列作为其特例的较为广泛的相依序列,但不同于NOD序列.利用LNQD序列的基本性质,我们给出了若干指数型不等式,如Bernstein型不等式等,由此可进一步研究其完全收敛性和几乎处处收敛性.利用LNQD序列的矩不等式,我们得到了LNQD序列的大偏差定理和Hajek-Renyi型不等式,并由此给出LNQD序列的强收敛速度和上确界的可积性.这些都是LNQD序列的新结果.
本文在最后一章主要研究弱(下)鞅及其凸函数的极值不等式和极限定理.弱鞅概念是在20世纪八十年代被提出来的,包含鞅作为其特例,并且均值为零的独立序列、PA序列和强正相依序列的部分和序列也是弱鞅.在本章我们做出了如下五方面的贡献：
·研究了弱(下)鞅及其凸函数的Chow型极大值不等式和Doob型极大值不等式,并且获得了一些概率不等式的新结果；
·建立了弱(下)鞅及其凸函数在0＜p＜1,p=1以及p＞1场合下的Doob型不等式,推广了文献Christofides和Wang中的相应结果；
·指出文献Harremoes中定理4的一个证明错误,并给出完整证明,同时我们还给出了弱鞅极小值不等式的一个新结果,由此证明了其中{Sn,n≥1}为一非负弱鞅,S1=1,γ(x)=x-1-log x,x>0上述结果推广了文献Harremoes中有关鞅的相应结果；
·给出了弱(下)鞅及其凸函数的一些新的强大数定律和强收敛速度,推广并改进了文献Christofides,Chow和Prakasa Rao中的相应结果；
·给出弱下鞅一致可积的一个等价条件.
The limit theory of dependent suquences has wide applications in applied probability, statistics, insurance and financial mathematics, complex systems, reliability theory, sur-vival analysis and other fields. This thesis focuses mainly on probability inequalities for demimartingale and three dependent sequences, such as Bernstein-type inequality, Hajek-Renyi-type inequality, Chow-type maximal inequality, Doob-type maximal inequality, and so forth. The limit theorems for demimartingale and three dependent sequences, such as complete convergence, almost sure convergence, strong growth rate, and so on, are also considered.
In Chapter 2, we study the moment inequality forφmixing suquence. By the moment inequality, we obtain the Kolmogorov-type convergence theorem, three series theorem, strong law of large numbers and strong growth rate forφmixing suquence. We also give some new results of probability inequalities forφmixing suquence, such as Hajek-Renyi-type inequality, and so on. As a consequence, the integrability of supremum forφmixing suquence can be proved. By the Bernstein-type inequality, we investigate the complete convergence and asymptotic approximation of inverse moment forφmixing suquence. The asymptotic approximation of inverse moment that we obtain generalizes and improves the results of Theorem 3 in Kaluszka and Okolewski, Theorems 2.1 and 2.3 in Hu et al. and Theorem 1 in Wu et al.(the condition sup1≤i≤n EZi/Bn≤C1 can be removed). In addition, we point out that there ia a mistake in the proof of the main results in Sun and Ling, and give the Bahadur representation for sample quantiles underφmixing sequence. We get better bound than that in Ling.
In Chapter 3, we investigate the exponential inequalities and strong convergence properties for bounded and unbounded NOD sequence. NOD sequence is a very broad class of dependent sequences, which contains independent sequence and NA sequence as special cases. The main purpose of this chapter is to establish the following exponential inequality for unbounded NOD sequence under suitable moment conditions: where is a sequence of positive numbers satisfying Furthermore, we study the strong convergence properties for NOD sequence, which gener-alize and improve the corresponding results for NA sequence in Kim and Kim, Nooghabi and Azarnoosh and Xing et al.
The probability inequalities and strong growth rate for LNQD sequence is the main target of Chapter 4. LNQD sequence is also a very broad class of dependent sequences, which contains independent sequence and NA sequence as special cases, but different from NOD sequence. By the Basic properties of LNQD sequence, we get some exponential inequalities, such as Bernstein-type inequality, and so forth, which can be applied to study the complete convergence and almost sure convergence. We also obtain the large deviations and Hajek-Renyi-type inequality from the moment inequality of LNQD suequence, which can be applied to prove the strong growth rate and the integrability of supremum. The results listed above are new results for LNQD suquence.
In the lase chapter of this thesis, we study the maximal inequalities and limit the-orem for demimartingale and its convex function. The concept of demimartingale was introduced in 1980s, which contains martingale as a special case. The partial sums of sequences of mean zero independent random variables, mean zero associated random vari-ables and mean zero strongly positive dependent random variables are all demimartingale. We make the following five contributions in this chapter:
●We study the Chow-type maximal inequality and Doob-type maximal inequality for demi(sub)martingale and its convex function. Some new results of probability inequalities are also obtained;
●We establish the Doob-type inequality for demi(sub)martingale and its convex function under the case of 0
●We point out that there is a mistake in the proof Theorem 4 in Harremoes, and give a complete proof. A new minimal inequality for demimartinge is also obtained, which can be applied to prove the folowing inequality: where {Sn,n≥1} is a nonnegative demimartingale, S1=1,γ(x)=x-1-logx,x>0. The result above generalizes the corresponding one of Harremoes;
●We give some new kinds of strong law of large numbers and strong growth rate for demi(sub)martingale and its convex function, which generalize and improve the corre-sponding ones of Christofides, Chow and Prakasa Rao;
●An equivalent condition of uniform integrability for demisubmartingales is pre-sented.
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