相依随机变量的强收敛性
摘要
本文是我在硕士阶段完成的，主要研究了相依随机变量的强大数定律和完全收敛性。全文共分三章：
第一章两两PQD序列的强大数律和完全收敛性
两两PQD序列的概念是Lehmann提出的，它是一类比较广泛的随机变量序列，包含常见的相伴序列。对这方面的研究已经有了一些结果，但不如相伴序列多。Matula得到了两两PQD序列几乎处处中心极限定理，严继高等给出了两两PQD序列的Jamison型加权部分和的强稳定性，陆凤彬给出了与PA列类似的完全收敛性。
在第一章中，我们主要考虑了两两PQD序列的Marcinkiewicz型强大数律和另一形式完全收敛性，得到如下结果：
定理0.1设{X_n；n≥1}是均值为0的两两PQD列，且sum from i=1 to∞v~(1/2)(2~i)＜∞。令1≤p＜2；φ：R→R~+是非负连续的偶函数，(?)φ(x)=∞，对某个1＜s≤2使得(?)↑且φ(x)/x~s↓，x→∞。如果sum from n=1 to∞log~2 n[(?)Eφ(|X_i|/φ(n~(1/p))]~(1/2)＜∞则1/n~(1/p)sum from i=1 to n X_i→0，a.s. n→∞。
定理0.2设{X_n；n≥1}是两两PQD列，对(?)ε＞0，有sum from j=1 to∞V_(ar)X_j/j~2+sum from j≠k=1 to∞Cov(X_j，X_k)/j·k＜∞和sum from j≠k=1 to∞P(|sum from i=1 to j(X_i-EX_i)|≥jε，|sum from i=1 to k(X_i-EX_i)|≥kε)＜∞。则sum from n=1 to∞(logn)~(-2)P(|sum from i=1 to n(X_i-EX_i)|≥ε)＜∞。
第二章一类PA序列非单调函数的强大数律和完全收敛性
1984年Newman给出了如下定义：
定义0.1设f和f_1是定义在R~n上两实值函数，则f《f_1当且仅当f+f_1，f-f_1均为逐点单调不减。特别地，如果f《f_1，则f_1为逐点非降的。
设{X_n；n≥1}为PA序列，令
(ⅰ)Y_n=f_n(X_1，X_2…，X_n)
(ⅱ)(?)_n=(?)_n(X_2，X_2…，X_n) (0.0.1)
(ⅲ)f_n《(?)_n
(ⅳ)EY_n~2＜∞，E(?)_n~2＜∞，n∈N
当条件(ⅰ)-(ⅳ)成立时，我们记为Y_n《(?)_n。其中f_n，(?)_n为实函数，且仅依赖于有限个X_n。关于{Y_n；n≥1}的极限性质已经有了一些结果。Matula(2001)证明了{Y_n；n≥1}的强大数律和中心极限定理，Dewan和Rao(2006)年得到了Hajek-Renyi型不等式及推广的三级数定理。在第二章我们得到如下结果：
定理0.3{X_n；n≥1}为PA列，Y_n，(?)_n如(0.0.1)式定义，且Y_n《(?)_n。gn(x)是偶函数列，且在区间x＞0取正值不减，而且对每一个n满足下列条件之一：
(ⅰ)区间x＞0中，x/gn(x)不减；
(ⅱ)区间x＞0中，x/gn(x)，gn(x)/x~2都不增且EY_n=E(?)_n=0。此外{α_n；n≥1}是常数列，满足0＜α_n↑∞，且sum from n=1 to∞E_(gn)(Y_n)/gn(a_n)＜∞和sum from n=1 to∞(E_(gn)((?)_n)/gn(a_n)~(1/2)＜∞。则当n→∞时1/α_n sum from k=1 to n Y_k→0 a.s。
定理0.4{X_n；n≥1}为PA列，Y_n，(?)_n如(0.0.1)式定义，且Y_n《(?)_n。若对(?)ε＞0，sum from j=1 to∞V_(ar)(?)_j/j~2+sum from j≠k=1 to∞Cov((?)_j,(?))k)/j·k＜∞和sum from j≠k=1 to∞P(|sum from i=1 to j(Y_i-EY_i)|≥jε，|sum from i=1 to k(Y_i-EY_i)|≥kε)＜∞。则sum from n=1 to∞P(|sum from i=1 to n(Y_i-EY_i)|≥ε)＜∞。
第三章相依线性过程的完全收敛性
假设{X_i；-∞＜i＜∞}是随机变量序列，{α_i；-∞＜i＜∞}是绝对可加的实数序列，定义线性过程{Y_k；k≥1}：Y_k=sum from i=-∞to∞α_(i+k)X_i。(0.0.2)
我们称(0.0.2)式定义的线性过程为滑动平均过程。滑动平均过程是时间序列的重要研究对象，许多作者在适当的假设下得到了相应的极限性质。例如Burton和Dehling(1990)在条件Eexp(tX_1)＜∞下得到了大偏差原理；Ibragimov(1962)建立了的中心极限定理；Yang(1996)得到重对数律；Li et al．(1992)得到完全收敛性。
本章第二节{X_i；—∞＜i＜∞}为两两NQD序列时的上述形式的完全收敛性。得到如下结果：
定理0.5设1＜p＜2，1/2＜α≤1，αp≥1。若
(ⅰ)h(x)＞0(x＞0)为→∞时的缓变函数；
(ⅱ){α_i；—∞＜i＜∞}是绝对可加的实数序列；
(ⅲ)EX_1=0，EX_1~2＜∞。则E|X_1|~ph(|X_1~(1/α)|)＜∞蕴涵sum from n=1 to∞n~(αp-2)h(n)P(|sum from k=1 to n Y_k|≥n~αε)＜∞，(?)ε＞0。
设{Y_t；t∈Z~+}是概率空间(Ω，(?)，P)上如下定义的线性过程：Y_t=sum from j=0 to∞α_jX_(t-j)。(0.0.3)其中{α_j；j≥0}是实数列，sum from j=0 to∞|α_j|＜∞；{X_t；t∈Z~+}是随机过程且EX_t=0，0＜EX_t~2＜∞。
我们称(0.0.3)式定义的线性过程为一般线性过程。显然，滑动平均过程是一般线性过程的特殊情况。此类线性过程在时间序列分析中起着特殊的作用，而且由于它在经济，工程，物理科学等方面的广泛应用而备受关注。大量的结果都是通过对{X_t；t∈Z~+}施加各种各样的条件得到的。Fakhre Zaberi和Lee建立了独立同分布时的CLT；1997年他们又证明了强混合条件下的FCLT；Tae-Sun Kim和Baek建立了平稳LPQD过程时的CLT；而Tae-Sun Kim，Mi-Hwa Ko和Dong Ho Park得到了LPQD和PA过程时的强大数定律。
本章第三节我们考虑了正相依样本下的线性过程的完全收敛性，得到如下结果：
定理0.6设1＜p＜2，1/2＜α≤1，αp≥1。若
(ⅰ)h(x)＞0(x＞0)为x→∞时的缓变函数；
(ⅱ){α_j；j≥0}是实数列且sum from j=0 to∞|α_j|＜∞；
(ⅲ){X_n；n≥1}是被X_0所界的PA列，EX_n=0且sum from i=1 to∞v~(1/2)(2~i)＜∞。
则E|X_0|~ph(|X_0~(1/α)|)＜∞蕴涵sum from n=1 to∞n~(αp-2)h(n)P(|S_n|≥εn~α)＜∞，(?)ε＞0。同时，对于两两PQD样本的一般线性过程，我们也得到类似定理0.6的结果。
This thesis is finished during my master of science, mainly discusses SLLN andcomplete convergence for dependent random variables. It consist of three chapters:
In chapterⅠ, we mainly discuss the strong law of large numbers and completeconvergence for pairwise PQD sequences.
The concept of pairwise PQD sequences was introduced by Lehmann. It is a kindof generic sequences, including associated sequences. Matula established a almost ev-erywhere central limit theorem; Yan Jigao proved a strong convergence for Jamison-wise weighted sums of pairwise PQD sequences; Lu Fengbin obtained a completeconvergence which is similar to PA sequences. In this chapter, the following resultsare obtained:
Theorem0.1 Suppose{X_n; n≥1}is a pairwise PQD sequence with mean 0, andsum from i=1 to∞v~(1/2)(2~i)＜∞. Let 1≤p＜2;φ: R→R~+ is nonnegative, even and continuousfunction, (?)φ(x)=∞, for some 1＜s≤2,φ(x)/x↑andφ(x)/x~8↓, x→∞. Assume sum from n=1 to∞log~2 n[(?) Eφ(|X_i|)/φ(n~(1/p)]~(1/2)＜∞Then 1/n~(1/p) sum from i=1 to n X_i→0, a.s. n→∞.
Theorem0.2 Suppose{X_n; n≥1} is a pairwise PQD sequence, for(?)ε＞0, sum from j=1 to∞VarX_j/j~2+sum from j≠k=1 to∞Cov(X_j, K_k)/j·k＜∞and sum from j≠k=1 to∞P(|sum from i=1 to j(X_i-EX_i)|≥jε, |sum from i=1 to k(X_i-EX_i)|≥kε)＜∞.Then sum from n=1 to∞(logn)~(-2)P(|sum from i=1 to n(X_i-EX_i)|≥ε)＜∞.
In chapterⅡ, we mainly discussed the SLLN for a sequence of nonmonotonic func-tions of associated random variables.
The following definition was given by Newman in 1984:
Definition0.1 Let f and f_1 be two real-valued functions defined on R~n, then f＜＜f_1 if and only if f+f_1, f-f_1 are both nondecreasing componentwise. In particular, if f＜＜f_1, then f_1 will be nondecreasing componentwise.
Let{X_n; n≥1} be a PA sequences. Let
(ⅰ) Y_n=f_n(X_1, X_2,…, X_n)
(ⅱ) (?)_n=(?)_n(X_1, X_2,…, X_n) (0.0.4)
(ⅲ) f_n＜＜(?)_n
(ⅳ) EY_n~2＜∞, E(?)_n~2＜∞, n∈N
If the conditions (ⅰ)-(ⅳ) hold, we write Y_n＜＜(?)_n. There are some limiting resultson{Y_n; n≥1}. Matula(2001) proved SLLN and CLT for {Y_n; n≥1}, Dewan andRao(2006)obtained Hajek-Renyi-tpye inequlity. We prove the following result:
Theorem0.3 {X_n; n≥1} is a PA sequence, Y_n, (?)_n is defined in (0.0.1), andY_n＜＜(?)_n. g_n(X) is even functions, and is positive and nondecreasing when x＞0, forevery n satisfies the alternative assumption that:
(ⅰ) x/g_n(x) is nondecreasing in (0,∞);
(ⅱ) x/g_n(x), g_n(x)/x~2 are nonincreasing in (0,∞). Meanwhile EY_n=E(?)_n=0.In addition, {a_n; n≥1} is a sequence of real numbers, with 0＜a_n↑∞, sum from n=1 to∞Eg_n(Y_n)/g_n(a_n)＜∞and sum from n=1 to∞(Eg_n((?)_n))/g_n(a_n)~(1/2)＜∞. Then when n→∞, 1/a_n sum from k=1 to n Y_k→0 a.s.
In addition, we prove a complete convergence for {Y_n; n≥1} similar to Theo-rem0.2.
In chapterⅢ, a complete convergence for linear processes under dependent as-sumption is discussed.
Assume that {X_i; -∞＜i＜∞} is a doubly infinite sequence, let {a_i; -∞＜i＜∞} be an absolutely summable sequence of real numbers, and {Y_k; k≥1}: Y_k=sum from i=-∞to∞a_(i+k)X_i. (0.0.5)
Linear processes defined as (0.0.5) are called moving average processes. Manylimiting results were obtained for moving average processes {Y_k; k≥1}. Burton andDehling(1990) obtained large deviation principle assuming Eexp(tX_1)＜∞; Ibragimov(1962) established CLT; Li et al.(1992) obtained a complete convergence. In sectionⅡ, weprove a complete convergence when{X_i; -∞＜i＜∞} is a pairwise NQD sequence:
Theorem0.5 Supposel＜p＜2, 1/2＜α＜1,αp≥1. Let
(ⅰ) h(x)＞0(x＞0) be a slowly varying function, when x→∞;
(ⅱ) {a_i; -∞＜i＜∞} be an absolutely summable sequence of real numbers;
(ⅲ) EX_1=0, EX_1~2＜∞. ThenE|X_1|~ph(|X_1~(1/α)|)＜∞imply sum from n=1 to∞n~(αp-2)h(n)P(|sum from k=1 to n Y_k|≥n~αε)＜∞, (?)ε＞0.
Let{Y_t; t∈Z~+} be a linear process defined on a probability space (Ω, (?), P): Y_t=sum from j=0 to∞a_jX_(t-j). (0.0.6)where {a_j; j≥0} is a sequence of real numbers, sum from j=0 to∞|a_j|＜∞; {X_t; t∈Z~+} is a processand EX_t=0, 0＜EX_t~2＜∞.
Linear processes defined as (0.0.6) are called general linear processes. Obviously, moving average processes are special cases of general linear processes. The linear pro-cesses are of special importance in time series analysis. Fakhre Zaberi and Lee obtainedCLT under i.i.d assumption; in 1997, they obtained FCLT under the strong mixingcondition; Tae-Sun Kim, Mi-Hwa Ko and Dong Ho Park proved SLLN under theLPQD and PA condition on {X_t; t∈Z~+}. In sectionⅢ, we discuss the linear pro-cess under positive dependence condition on {X_t; t∈Z~+}, and obtain the followingresult:
Therorem0.6 Supposel＜p＜2, 1/2＜α≤1,αp≥1. Let
(ⅰ) h(x)＞0(x＞0) be a slowly varying function, when x→∞;
(ⅱ) {a_j; j≥0} be a sequence of real numbers and sum from j=0 to∞|a_j|＜∞;
(ⅲ) {X_n; n≥1} is a PA sequence bounded by X_0, EX_n =0 and sum from i=1 to∞v~(1/2)(2~i)＜∞.
Then E|X_0|~Ph(|X_0~(1/α)|)＜∞imply sum from n=1 to∞n~(αp-2)h(n)P(|S_n|≥εn~α)＜∞, (?)ε＞0.
Meanwhile, we obtain a result under pairwise PQD condition on {X_t; t∈Z~+}similar to the above.
第一章两两PQD序列的强大数律和完全收敛性
两两PQD序列的概念是Lehmann提出的，它是一类比较广泛的随机变量序列，包含常见的相伴序列。对这方面的研究已经有了一些结果，但不如相伴序列多。Matula得到了两两PQD序列几乎处处中心极限定理，严继高等给出了两两PQD序列的Jamison型加权部分和的强稳定性，陆凤彬给出了与PA列类似的完全收敛性。
在第一章中，我们主要考虑了两两PQD序列的Marcinkiewicz型强大数律和另一形式完全收敛性，得到如下结果：
定理0.1设{X_n；n≥1}是均值为0的两两PQD列，且sum from i=1 to∞v~(1/2)(2~i)＜∞。令1≤p＜2；φ：R→R~+是非负连续的偶函数，(?)φ(x)=∞，对某个1＜s≤2使得(?)↑且φ(x)/x~s↓，x→∞。如果sum from n=1 to∞log~2 n[(?)Eφ(|X_i|/φ(n~(1/p))]~(1/2)＜∞则1/n~(1/p)sum from i=1 to n X_i→0，a.s. n→∞。
定理0.2设{X_n；n≥1}是两两PQD列，对(?)ε＞0，有sum from j=1 to∞V_(ar)X_j/j~2+sum from j≠k=1 to∞Cov(X_j，X_k)/j·k＜∞和sum from j≠k=1 to∞P(|sum from i=1 to j(X_i-EX_i)|≥jε，|sum from i=1 to k(X_i-EX_i)|≥kε)＜∞。则sum from n=1 to∞(logn)~(-2)P(|sum from i=1 to n(X_i-EX_i)|≥ε)＜∞。
第二章一类PA序列非单调函数的强大数律和完全收敛性
1984年Newman给出了如下定义：
定义0.1设f和f_1是定义在R~n上两实值函数，则f《f_1当且仅当f+f_1，f-f_1均为逐点单调不减。特别地，如果f《f_1，则f_1为逐点非降的。
设{X_n；n≥1}为PA序列，令
(ⅰ)Y_n=f_n(X_1，X_2…，X_n)
(ⅱ)(?)_n=(?)_n(X_2，X_2…，X_n) (0.0.1)
(ⅲ)f_n《(?)_n
(ⅳ)EY_n~2＜∞，E(?)_n~2＜∞，n∈N
当条件(ⅰ)-(ⅳ)成立时，我们记为Y_n《(?)_n。其中f_n，(?)_n为实函数，且仅依赖于有限个X_n。关于{Y_n；n≥1}的极限性质已经有了一些结果。Matula(2001)证明了{Y_n；n≥1}的强大数律和中心极限定理，Dewan和Rao(2006)年得到了Hajek-Renyi型不等式及推广的三级数定理。在第二章我们得到如下结果：
定理0.3{X_n；n≥1}为PA列，Y_n，(?)_n如(0.0.1)式定义，且Y_n《(?)_n。gn(x)是偶函数列，且在区间x＞0取正值不减，而且对每一个n满足下列条件之一：
(ⅰ)区间x＞0中，x/gn(x)不减；
(ⅱ)区间x＞0中，x/gn(x)，gn(x)/x~2都不增且EY_n=E(?)_n=0。此外{α_n；n≥1}是常数列，满足0＜α_n↑∞，且sum from n=1 to∞E_(gn)(Y_n)/gn(a_n)＜∞和sum from n=1 to∞(E_(gn)((?)_n)/gn(a_n)~(1/2)＜∞。则当n→∞时1/α_n sum from k=1 to n Y_k→0 a.s。
定理0.4{X_n；n≥1}为PA列，Y_n，(?)_n如(0.0.1)式定义，且Y_n《(?)_n。若对(?)ε＞0，sum from j=1 to∞V_(ar)(?)_j/j~2+sum from j≠k=1 to∞Cov((?)_j,(?))k)/j·k＜∞和sum from j≠k=1 to∞P(|sum from i=1 to j(Y_i-EY_i)|≥jε，|sum from i=1 to k(Y_i-EY_i)|≥kε)＜∞。则sum from n=1 to∞P(|sum from i=1 to n(Y_i-EY_i)|≥ε)＜∞。
第三章相依线性过程的完全收敛性
假设{X_i；-∞＜i＜∞}是随机变量序列，{α_i；-∞＜i＜∞}是绝对可加的实数序列，定义线性过程{Y_k；k≥1}：Y_k=sum from i=-∞to∞α_(i+k)X_i。(0.0.2)
我们称(0.0.2)式定义的线性过程为滑动平均过程。滑动平均过程是时间序列的重要研究对象，许多作者在适当的假设下得到了相应的极限性质。例如Burton和Dehling(1990)在条件Eexp(tX_1)＜∞下得到了大偏差原理；Ibragimov(1962)建立了的中心极限定理；Yang(1996)得到重对数律；Li et al．(1992)得到完全收敛性。
本章第二节{X_i；—∞＜i＜∞}为两两NQD序列时的上述形式的完全收敛性。得到如下结果：
定理0.5设1＜p＜2，1/2＜α≤1，αp≥1。若
(ⅰ)h(x)＞0(x＞0)为→∞时的缓变函数；
(ⅱ){α_i；—∞＜i＜∞}是绝对可加的实数序列；
(ⅲ)EX_1=0，EX_1~2＜∞。则E|X_1|~ph(|X_1~(1/α)|)＜∞蕴涵sum from n=1 to∞n~(αp-2)h(n)P(|sum from k=1 to n Y_k|≥n~αε)＜∞，(?)ε＞0。
设{Y_t；t∈Z~+}是概率空间(Ω，(?)，P)上如下定义的线性过程：Y_t=sum from j=0 to∞α_jX_(t-j)。(0.0.3)其中{α_j；j≥0}是实数列，sum from j=0 to∞|α_j|＜∞；{X_t；t∈Z~+}是随机过程且EX_t=0，0＜EX_t~2＜∞。
我们称(0.0.3)式定义的线性过程为一般线性过程。显然，滑动平均过程是一般线性过程的特殊情况。此类线性过程在时间序列分析中起着特殊的作用，而且由于它在经济，工程，物理科学等方面的广泛应用而备受关注。大量的结果都是通过对{X_t；t∈Z~+}施加各种各样的条件得到的。Fakhre Zaberi和Lee建立了独立同分布时的CLT；1997年他们又证明了强混合条件下的FCLT；Tae-Sun Kim和Baek建立了平稳LPQD过程时的CLT；而Tae-Sun Kim，Mi-Hwa Ko和Dong Ho Park得到了LPQD和PA过程时的强大数定律。
本章第三节我们考虑了正相依样本下的线性过程的完全收敛性，得到如下结果：
定理0.6设1＜p＜2，1/2＜α≤1，αp≥1。若
(ⅰ)h(x)＞0(x＞0)为x→∞时的缓变函数；
(ⅱ){α_j；j≥0}是实数列且sum from j=0 to∞|α_j|＜∞；
(ⅲ){X_n；n≥1}是被X_0所界的PA列，EX_n=0且sum from i=1 to∞v~(1/2)(2~i)＜∞。
则E|X_0|~ph(|X_0~(1/α)|)＜∞蕴涵sum from n=1 to∞n~(αp-2)h(n)P(|S_n|≥εn~α)＜∞，(?)ε＞0。同时，对于两两PQD样本的一般线性过程，我们也得到类似定理0.6的结果。
This thesis is finished during my master of science, mainly discusses SLLN andcomplete convergence for dependent random variables. It consist of three chapters:
In chapterⅠ, we mainly discuss the strong law of large numbers and completeconvergence for pairwise PQD sequences.
The concept of pairwise PQD sequences was introduced by Lehmann. It is a kindof generic sequences, including associated sequences. Matula established a almost ev-erywhere central limit theorem; Yan Jigao proved a strong convergence for Jamison-wise weighted sums of pairwise PQD sequences; Lu Fengbin obtained a completeconvergence which is similar to PA sequences. In this chapter, the following resultsare obtained:
Theorem0.1 Suppose{X_n; n≥1}is a pairwise PQD sequence with mean 0, andsum from i=1 to∞v~(1/2)(2~i)＜∞. Let 1≤p＜2;φ: R→R~+ is nonnegative, even and continuousfunction, (?)φ(x)=∞, for some 1＜s≤2,φ(x)/x↑andφ(x)/x~8↓, x→∞. Assume sum from n=1 to∞log~2 n[(?) Eφ(|X_i|)/φ(n~(1/p)]~(1/2)＜∞Then 1/n~(1/p) sum from i=1 to n X_i→0, a.s. n→∞.
Theorem0.2 Suppose{X_n; n≥1} is a pairwise PQD sequence, for(?)ε＞0, sum from j=1 to∞VarX_j/j~2+sum from j≠k=1 to∞Cov(X_j, K_k)/j·k＜∞and sum from j≠k=1 to∞P(|sum from i=1 to j(X_i-EX_i)|≥jε, |sum from i=1 to k(X_i-EX_i)|≥kε)＜∞.Then sum from n=1 to∞(logn)~(-2)P(|sum from i=1 to n(X_i-EX_i)|≥ε)＜∞.
In chapterⅡ, we mainly discussed the SLLN for a sequence of nonmonotonic func-tions of associated random variables.
The following definition was given by Newman in 1984:
Definition0.1 Let f and f_1 be two real-valued functions defined on R~n, then f＜＜f_1 if and only if f+f_1, f-f_1 are both nondecreasing componentwise. In particular, if f＜＜f_1, then f_1 will be nondecreasing componentwise.
Let{X_n; n≥1} be a PA sequences. Let
(ⅰ) Y_n=f_n(X_1, X_2,…, X_n)
(ⅱ) (?)_n=(?)_n(X_1, X_2,…, X_n) (0.0.4)
(ⅲ) f_n＜＜(?)_n
(ⅳ) EY_n~2＜∞, E(?)_n~2＜∞, n∈N
If the conditions (ⅰ)-(ⅳ) hold, we write Y_n＜＜(?)_n. There are some limiting resultson{Y_n; n≥1}. Matula(2001) proved SLLN and CLT for {Y_n; n≥1}, Dewan andRao(2006)obtained Hajek-Renyi-tpye inequlity. We prove the following result:
Theorem0.3 {X_n; n≥1} is a PA sequence, Y_n, (?)_n is defined in (0.0.1), andY_n＜＜(?)_n. g_n(X) is even functions, and is positive and nondecreasing when x＞0, forevery n satisfies the alternative assumption that:
(ⅰ) x/g_n(x) is nondecreasing in (0,∞);
(ⅱ) x/g_n(x), g_n(x)/x~2 are nonincreasing in (0,∞). Meanwhile EY_n=E(?)_n=0.In addition, {a_n; n≥1} is a sequence of real numbers, with 0＜a_n↑∞, sum from n=1 to∞Eg_n(Y_n)/g_n(a_n)＜∞and sum from n=1 to∞(Eg_n((?)_n))/g_n(a_n)~(1/2)＜∞. Then when n→∞, 1/a_n sum from k=1 to n Y_k→0 a.s.
In addition, we prove a complete convergence for {Y_n; n≥1} similar to Theo-rem0.2.
In chapterⅢ, a complete convergence for linear processes under dependent as-sumption is discussed.
Assume that {X_i; -∞＜i＜∞} is a doubly infinite sequence, let {a_i; -∞＜i＜∞} be an absolutely summable sequence of real numbers, and {Y_k; k≥1}: Y_k=sum from i=-∞to∞a_(i+k)X_i. (0.0.5)
Linear processes defined as (0.0.5) are called moving average processes. Manylimiting results were obtained for moving average processes {Y_k; k≥1}. Burton andDehling(1990) obtained large deviation principle assuming Eexp(tX_1)＜∞; Ibragimov(1962) established CLT; Li et al.(1992) obtained a complete convergence. In sectionⅡ, weprove a complete convergence when{X_i; -∞＜i＜∞} is a pairwise NQD sequence:
Theorem0.5 Supposel＜p＜2, 1/2＜α＜1,αp≥1. Let
(ⅰ) h(x)＞0(x＞0) be a slowly varying function, when x→∞;
(ⅱ) {a_i; -∞＜i＜∞} be an absolutely summable sequence of real numbers;
(ⅲ) EX_1=0, EX_1~2＜∞. ThenE|X_1|~ph(|X_1~(1/α)|)＜∞imply sum from n=1 to∞n~(αp-2)h(n)P(|sum from k=1 to n Y_k|≥n~αε)＜∞, (?)ε＞0.
Let{Y_t; t∈Z~+} be a linear process defined on a probability space (Ω, (?), P): Y_t=sum from j=0 to∞a_jX_(t-j). (0.0.6)where {a_j; j≥0} is a sequence of real numbers, sum from j=0 to∞|a_j|＜∞; {X_t; t∈Z~+} is a processand EX_t=0, 0＜EX_t~2＜∞.
Linear processes defined as (0.0.6) are called general linear processes. Obviously, moving average processes are special cases of general linear processes. The linear pro-cesses are of special importance in time series analysis. Fakhre Zaberi and Lee obtainedCLT under i.i.d assumption; in 1997, they obtained FCLT under the strong mixingcondition; Tae-Sun Kim, Mi-Hwa Ko and Dong Ho Park proved SLLN under theLPQD and PA condition on {X_t; t∈Z~+}. In sectionⅢ, we discuss the linear pro-cess under positive dependence condition on {X_t; t∈Z~+}, and obtain the followingresult:
Therorem0.6 Supposel＜p＜2, 1/2＜α≤1,αp≥1. Let
(ⅰ) h(x)＞0(x＞0) be a slowly varying function, when x→∞;
(ⅱ) {a_j; j≥0} be a sequence of real numbers and sum from j=0 to∞|a_j|＜∞;
(ⅲ) {X_n; n≥1} is a PA sequence bounded by X_0, EX_n =0 and sum from i=1 to∞v~(1/2)(2~i)＜∞.
Then E|X_0|~Ph(|X_0~(1/α)|)＜∞imply sum from n=1 to∞n~(αp-2)h(n)P(|S_n|≥εn~α)＜∞, (?)ε＞0.
Meanwhile, we obtain a result under pairwise PQD condition on {X_t; t∈Z~+}similar to the above.
引文
[1] Hsu, P. L. and Robbins, H. Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. 33, 25-31, 1947.
[2] Lehmann, E. L.. Some concepts of dependence. Ann. Math. Statistics. 43, 1137-1153, 1966.
[3] 严继高，王岳宝等．关于两两PQD列的加权乘积和的强稳定性及乘积和的Marcinkiewicz型强大数律．应用概率统计．18，27-33，2002．
[4] 陆凤彬．两两尸QD序列的完全收敛性和强大数律．应用数学．16，29-33，2003．
[5] 林正炎，陆传荣，苏中根．概率极限理论基础．北京，高等教育出版社，1999．
[6] Fazekas, I. and Klesov, O.. A general approach to the srong laws of large numbers. Teor. Verojatnost. i Primenen. 45, 569-583, 2000.
[7] Kuczmaszewska, A.. The strong law of large numbers for dependent random variables. Statist. Probab. Lett. 73, 305-314, 2005.
[8] New, C. M.. Asymptotic independence and limit theorems for positively and negatively dependent random variables. Inequalities in Statistics and Probability, IMS, Hayward, 127-140, 1984.
[9] Dewan, I. and Prakasa Rao, B. L. S.. Asymptotic normality for U-statistics of associated random variables. Journal of Statistical Planning and Inference. 97, 201-225, 2001.
[10] Matula, P.. Limit theorems for sums of nonmonotonic functions of associated random variables. Journal of Mathenatical Science. 105, 2590-2593, 2001.
[11] Dewan, I. and Prakasa Rao, B. L. S.. Hajek-Renyi-type inequality for some nonmonotonic functions of associated random variables. Journal of Inequalities and Applications. 1-8, 2006.
[12] 吴群英．混合序列的概率极限理论．北京，科学出版社，2006．
[13] zhang, Li-xin. Complete convergence of moving average process under dependent assumptions. Statist. Probab. Lett. 30, 165-170, 1996.
[14] Fakhre-Zakeri, I. and lee, S. Sequential estimation of the mean of a linear process. Sequential Anal. 11, 181-197, 1992.
[15] Fakhre-Zakeri, I. and lee, S.. A random functional central limit theorem for stationary linear processes generated by matingales. Statist. Probab. Lett. 35, 417-422, 1997.
[16] Kim Tae-Sung and Baek Jong-Il. A central limit theorem for stationary linear processes generated by linear positively quadrant-dependent process. Statist. Probab. Lett. 51, 299-305, 2001.
[17] Kim Tae-Sung, Ko Mi-Hwa and Park Dong-Ho. Almost sure convergence for linear processes generated by positively dependent processes. Stochastic Analysis and Applications. 22, 143-153, 2004.
[18] Sung, Soo Hak. Complete convergence for weighted sums of random variables. Statist. Probab. Lett. 77, 303-311, 2007.
[19] Burton, R. M. and Dehling, H.. Large deviations for some weakly dependent random process. Statist. Probab. Lett. 9, 397-401, 1990.
[20] 杨善朝．PA序列部分和的完全收敛性．应用概率统计．17，197-202，2001．
[2] Lehmann, E. L.. Some concepts of dependence. Ann. Math. Statistics. 43, 1137-1153, 1966.
[3] 严继高，王岳宝等．关于两两PQD列的加权乘积和的强稳定性及乘积和的Marcinkiewicz型强大数律．应用概率统计．18，27-33，2002．
[4] 陆凤彬．两两尸QD序列的完全收敛性和强大数律．应用数学．16，29-33，2003．
[5] 林正炎，陆传荣，苏中根．概率极限理论基础．北京，高等教育出版社，1999．
[6] Fazekas, I. and Klesov, O.. A general approach to the srong laws of large numbers. Teor. Verojatnost. i Primenen. 45, 569-583, 2000.
[7] Kuczmaszewska, A.. The strong law of large numbers for dependent random variables. Statist. Probab. Lett. 73, 305-314, 2005.
[8] New, C. M.. Asymptotic independence and limit theorems for positively and negatively dependent random variables. Inequalities in Statistics and Probability, IMS, Hayward, 127-140, 1984.
[9] Dewan, I. and Prakasa Rao, B. L. S.. Asymptotic normality for U-statistics of associated random variables. Journal of Statistical Planning and Inference. 97, 201-225, 2001.
[10] Matula, P.. Limit theorems for sums of nonmonotonic functions of associated random variables. Journal of Mathenatical Science. 105, 2590-2593, 2001.
[11] Dewan, I. and Prakasa Rao, B. L. S.. Hajek-Renyi-type inequality for some nonmonotonic functions of associated random variables. Journal of Inequalities and Applications. 1-8, 2006.
[12] 吴群英．混合序列的概率极限理论．北京，科学出版社，2006．
[13] zhang, Li-xin. Complete convergence of moving average process under dependent assumptions. Statist. Probab. Lett. 30, 165-170, 1996.
[14] Fakhre-Zakeri, I. and lee, S. Sequential estimation of the mean of a linear process. Sequential Anal. 11, 181-197, 1992.
[15] Fakhre-Zakeri, I. and lee, S.. A random functional central limit theorem for stationary linear processes generated by matingales. Statist. Probab. Lett. 35, 417-422, 1997.
[16] Kim Tae-Sung and Baek Jong-Il. A central limit theorem for stationary linear processes generated by linear positively quadrant-dependent process. Statist. Probab. Lett. 51, 299-305, 2001.
[17] Kim Tae-Sung, Ko Mi-Hwa and Park Dong-Ho. Almost sure convergence for linear processes generated by positively dependent processes. Stochastic Analysis and Applications. 22, 143-153, 2004.
[18] Sung, Soo Hak. Complete convergence for weighted sums of random variables. Statist. Probab. Lett. 77, 303-311, 2007.
[19] Burton, R. M. and Dehling, H.. Large deviations for some weakly dependent random process. Statist. Probab. Lett. 9, 397-401, 1990.
[20] 杨善朝．PA序列部分和的完全收敛性．应用概率统计．17，197-202，2001．