加權(quán)AANA隨機(jī)變量和的一致強(qiáng)收斂速度

胡宏昌

加權(quán)AANA隨機(jī)變量和的一致強(qiáng)收斂速度

胡宏昌

(湖北師范學(xué)院數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,湖北黃石435002)

為了完善AANA序列的極限理論,利用三級(jí)數(shù)定理、Borel-Cantelli引理及一些概率不等式,研究了AANA隨機(jī)變量序列的函數(shù)加權(quán)和.在一定的條件下,得到了其一致強(qiáng)收斂速度為n?13logn,推廣了關(guān)于NA隨機(jī)變量序列的相應(yīng)結(jié)果.

漸近幾乎負(fù)相關(guān);函數(shù)加權(quán)和;一致強(qiáng)收斂速度

1 引言

定義1.1[1-2]稱{ξi,i≥1}為漸近幾乎負(fù)相關(guān)(asymptotically almost negatively associated,簡(jiǎn)記為AANA)序列,如果方差存在,且存在非負(fù)序列q(i)→0(i→∞),使得

其中i,k≥1,f和g是坐標(biāo)方式為非遞減的連續(xù)函數(shù).

若令q(i)=0(i≥1),則AANA序列就是負(fù)相依[3](negatively associated,簡(jiǎn)記為NA)隨機(jī)序列,然而NA序列不一定就是AANA序列[1].因此,將有關(guān)NA序列的結(jié)果推廣到AANA序列的情形具有特別重要的理論意義.目前,關(guān)于AANA隨機(jī)變量已有一些結(jié)果.如:文獻(xiàn)[4]研究了強(qiáng)大數(shù)定律和重對(duì)數(shù)定律,文獻(xiàn)[5-6]分別得到了H′ajek-R′enyi不等式和Rosenthal型不等式,文獻(xiàn)[7]討論了部分和的極限,文獻(xiàn)[8]研究了極大不等式和強(qiáng)大數(shù)定律,文獻(xiàn)[9]研究了加權(quán)和的強(qiáng)大數(shù)定律.然而,有關(guān)AANA隨機(jī)變量序列的函數(shù)加權(quán)和的極限性質(zhì)還沒有人研究.因此,為了完善關(guān)于AANA序列的極限理論及研究其它有關(guān)隨機(jī)變量的極限的性質(zhì),本文研究AANA隨機(jī)變量序列的函數(shù)加權(quán)和的一致強(qiáng)收斂速度,推廣了關(guān)于獨(dú)立及NA隨機(jī)變量序列的相應(yīng)結(jié)果.

2 主要結(jié)果

定理2.1設(shè){ξi,i≥1}為AANA隨機(jī)序列,Eξi=0混合系數(shù)為{q(i),i≥1}.又設(shè){ani(t),1≤i≤n,n≥1}是定義在[0,1]上的函數(shù),且滿足

(1)對(duì)任意t∈[0,1],有

(2)對(duì)任意s,t∈[0,1],有

對(duì)于某個(gè)p>3,若

成立,則

由定理2.1很容易得到如下推論.該結(jié)論對(duì)研究誤差為NA序列的非線性、半?yún)?shù)等回歸模型的強(qiáng)收斂速度起著關(guān)鍵性的作用.

推論2.1[10]設(shè){ξi,i≥1}是均值為0的NA隨機(jī)序列,且函數(shù)

滿足定理2.1的條件.若對(duì)于某個(gè)p>3,

則

3 定理2.1的證明

為了證明定理2.1,先給出如下幾個(gè)引理.

引理3.1[10]對(duì)于隨機(jī)變量序列{ξi,i≥1},如果且存在常數(shù)M>0,使得|ξi|≤M,則對(duì)于任給的有

引理3.2[9]設(shè){ξi,i≥1}是混合系數(shù)為{q(i),i≥1}的AANA序列,且

如果對(duì)某個(gè)C>0,級(jí)數(shù)

引理3.3設(shè){ξi,i≥1}是混合系數(shù)為{q(i),i≥1}的AANA序列,且

?ε>0,若記

則{ξi1}和{ξi2}均為AANA序列,且<∞,a.s..

以下完成定理的證明.

將區(qū)間[0,1]分割成n2個(gè)小區(qū)間Ai=[si?1,si],i=1,2,···,n2,使得

于是對(duì)于任意的t∈[0,1],存在s使得s,t∈Ai.記

注意到Eξi=0,易得

由引理3.3及(1)式,得

由(2)式,得

同理可得,

由Cauchy-Schwarz和Markov不等式,以及,可得

下面證明:

令

則|Wi|≤M.又令

則由Chebyshev-Markov不等式和引理3.1,得

利用對(duì)稱性,易得

從而有

取C>3,則由(7)式得,

從而由(8)式及Borel-Cantelli引理即可得(6)式.綜上,由(1)-(6)式知本定理成立.

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Uniformly strong convergence rate for weighted sums of AANA random variables

Hu Hongchang
(School of Mathematics and Statistics,Hubei Normal University,Huangshi435002,China)

In order to improve the limit theory of the AANA sequence,by using the three series theorem, Borel-Cantelli lemma and some probability inequalities,we investigate function weighted sums of asymptotically almost negatively associated(AANA)random variables,and obtain uniformly strong convergence rate nlogn under some conditions.Hence the results generalize the corresponding one for negatively associated random variables.

asymptotically almost negatively associated,function weighted sums,uniformly strong, convergence rate

O211.4

A

1008-5513(2014)06-0558-06

10.3969/j.issn.1008-5513.2014.06.002

2014-06-13.

國(guó)家自然科學(xué)基金(11471105,11471223);湖北省教育廳青年資助項(xiàng)目(Q20142501).

胡宏昌(1971-),博士,教授,研究方向:回歸模型的統(tǒng)計(jì)推斷及其應(yīng)用,時(shí)間序列分析.

2010 MSC:60F15