關麗紅, 趙亞男

(長春大學 理學院, 長春 130022)

長程相依過程精確漸近性的一般結果

關麗紅, 趙亞男

(長春大學 理學院, 長春 130022)

長程相依過程; 矩完全收斂性; 精確漸近性; 一般結果; 分數(shù)積分過程

0 引言及主要結果

(H1) {ak,k≥0}為滿足條件ak～k-αl(k)的實數(shù)序列, 1/2<α<1;

(H3)g(x)為[n0,∞)上具有非負導數(shù)g′(x)的正值可導函數(shù), 且g(x)↑∞,x→∞;

本文的主要結果如下.

其中bn,N定義如定理1.

注1滿足假設條件(H2)的緩變函數(shù)有很多[16], 如l(x)=(logx)β, (loglogx)γ, elogδx, 其中:β,γ為實數(shù); 0<δ<1.

注2滿足假設條件(H3)～(H7)的g(x)有很多, 如g(x)=xα, (logx)β, (loglogx)γ, 其中α>0,β>0,γ>0為某些適當?shù)膮?shù).

注3在定理1中, 令s=1/2, 0≤p<2,g(x)=x, 則可得文獻[13]中的定理1.2; 令1/s=β(δ+1),p=2,g(x)=(logx)δ+1, 其中β≥2,δ>2/β-1, 則可得文獻[13]中的定理1.3.在定理2中,令p=2,s=1/2,g(x)=x, 則可得文獻[13]中的定理1.1, 因此本文推廣了長程相依過程的已有結果.

1 定理的證明

引理1[15]假設{Xt}為式(1)定義的滿足條件(H1)和(H2)的長程相依過程, 則

其中:bn如定理1所定義;W(s)為分數(shù)布朗運動.

其中bn如定理1所定義.

令A(ε)=[g-1(Mε-1/s)], 其中:g-1(x)為g(x)的反函數(shù);M≥1.

命題1在定理1的假設條件下, 有

證明: 利用引理1和引理2, 類似文獻[12]中命題1的證明可得.

命題2在定理1的假設條件下, 對于p>0, 有

證明: 類似文獻[8]中命題5.1的證明可得.

命題3在定理1的假設條件下, 對于p>0, 有

證明: 顯然

其中:

先估計Δn1.注意到n≤A(ε)即εgs(n)≤Ms, 則有

其次估計Δn3.注意到正態(tài)分布的任意階矩都存在, 則由Markov不等式, 有

最后估計Δn2.由Markov不等式和引理2, 并注意到q>1/s>p>0, 有

根據(jù)引理1, 當n→∞時,Δn→0, 因此由式(6)～(9), 可得

Δn1+Δn2+Δn3→0,n→∞.(10)

再由式(10)、φ(x)的單調性以及Toeplitz引理[24], 可知式(5)成立.證畢.

命題4在定理1的條件下, 對于p>0, 有

證明: 類似文獻[8]中命題5.3的證明可得.

命題5在定理1的條件下, 對于p>0, 有

證明: 注意到q>1/s>p>0, 由Markov不等式及引理2, 有

下面證明定理1.當p=0時, 由于

則由命題1可知定理1成立.當1/s>p>0, 注意到

故要證明式(2), 只需證明下列兩式成立即可:

由命題1可知式(13)成立, 由命題2～命題5以及三角不等式可知式(14)成立, 從而式(2)成立.

定理2的證明與定理1的證明類似.

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GeneralResultofPreciseAsymptoticsforLongMemoryProcesses

GUAN Lihong, ZHAO Yanan
(SchoolofScience,ChangchunUniversity,Changchun130022,China)

long memory process; complete moment convergence; precise asymptotics; general result; fractionally integrating process

2014-03-14.

關麗紅(1976—), 女, 滿族, 碩士, 講師, 從事概率統(tǒng)計與應用數(shù)學的研究, E-mail: guanlihong14@163.com.

國家自然科學基金(批準號: 11371085)和吉林省教育廳“十二五”科技研究項目(批準號: 吉教科合字[2014]第526號).

O211.4

A

1671-5489(2014)06-1191-05

10.13413/j.cnki.jdxblxb.2014.06.16

趙立芹)