長(zhǎng)程相依過程精確漸近性的一般結(jié)果

關(guān)麗紅, 趙亞男

(長(zhǎng)春大學(xué) 理學(xué)院, 長(zhǎng)春 130022)

長(zhǎng)程相依過程精確漸近性的一般結(jié)果

關(guān)麗紅, 趙亞男

(長(zhǎng)春大學(xué) 理學(xué)院, 長(zhǎng)春 130022)

長(zhǎng)程相依過程; 矩完全收斂性; 精確漸近性; 一般結(jié)果; 分?jǐn)?shù)積分過程

0 引言及主要結(jié)果

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

(H3)g(x)為[n0,∞)上具有非負(fù)導(dǎo)數(shù)g′(x)的正值可導(dǎo)函數(shù), 且g(x)↑∞,x→∞;

本文的主要結(jié)果如下.

其中bn,N定義如定理1.

注1滿足假設(shè)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(shí)數(shù); 0<δ<1.

注2滿足假設(shè)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(dāng)?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(xiàn)[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(xiàn)[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(xiàn)[13]中的定理1.1, 因此本文推廣了長(zhǎng)程相依過程的已有結(jié)果.

1 定理的證明

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

其中:bn如定理1所定義;W(s)為分?jǐn)?shù)布朗運(yùn)動(dòng).

其中bn如定理1所定義.

令A(yù)(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(shè)條件下, 有

證明: 利用引理1和引理2, 類似文獻(xiàn)[12]中命題1的證明可得.

命題2在定理1的假設(shè)條件下, 對(duì)于p>0, 有

證明: 類似文獻(xiàn)[8]中命題5.1的證明可得.

命題3在定理1的假設(shè)條件下, 對(duì)于p>0, 有

證明: 顯然

其中:

先估計(jì)Δn1.注意到n≤A(ε)即εgs(n)≤Ms, 則有

其次估計(jì)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(jì)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(dāng)n→∞時(shí),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(diào)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(duì)于p>0, 有

證明: 類似文獻(xiàn)[8]中命題5.3的證明可得.

命題5在定理1的條件下, 對(duì)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(dāng)p=0時(shí), 由于

則由命題1可知定理1成立.當(dāng)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.

[1]Hsu P L, Robbins H.Complete Convergence and the Strong Law of Large Numbers [J].Proc Nat Acad Sci USA, 1947, 33(2): 25-31.

[2]Heyde C C.A Supplement to the Strong Law of Large Numbers [J].J Appl Probab, 1975, 12: 173-175.

[3]Gut A, Sp?taru A.Precise Asymptotics in the Baum-Katz and Davis Laws of Large Numbers [J].J Math Anal Appl, 2000, 248(1): 233-246.

[4]WANG Yuebao, YANG Yang.A General Law of Precise Asymptotics for the Counting Process of Record Times [J].J Math Anal Appl, 2003, 286(2): 753-764.

[5]LIU Weidong, LIN Zhengyan.Precise Asymptotics for a New Kind of Complete Moment Convergence [J].Statist Probab Lett, 2006, 76(16): 1787-1799.

[6]劉君, 董志山, 張勇.由強(qiáng)混合序列生成線性過程精確漸近性的一般形式 [J].吉林大學(xué)學(xué)報(bào): 理學(xué)版, 2008, 46(4): 595-600.(LIU Jun, DONG Zhishan, ZHANG Yong.A General Result on Precise Asymptotics for the Linear Process Generated by Strong Mixing Sequences [J].Journal of Jilin University: Science Edition, 2008, 46(4): 595-600.)

[7]SU Zhonggen.Precise Asymptotics for Random Matrices and Random Growth Models [J].Acta Math Sin: Engl Ser, 2008, 24(6): 971-982.

[8]ZHANG Yong, YANG Xiaoyun, DONG Zhishan.A General Law of Precise Asymptotics for the Complete Moment Convergence [J].Chin Ann Math: Ser B, 2009, 30(1): 77-90.

[9]FU Ke’ang, ZHANG Lixin.Pecise Rates in the Law of the Logarithm for the Moment Convergence in Hilbert Spaces [J].Acta Math Sin: Engl Ser, 2007, 23(6): 965-972.

[10]TAN Xili, YANG Xiaoyun.The Precise Asymptotics of the Complete Convergence for Moving Average Processes ofm-DependentB-Valued Elements [J].Acta Math Sin: Engl Ser, 2009, 25(3): 467-480.

[11]ZHANG Yong, YANG Xiaoyun, DONG Zhishan.A General Law of Precise Asymptotics for Products of Sums under Dependence [J].Acta Math Sin: Engl Ser, 2010, 26(1): 107-116.

[12]吳素文, 張勇, 李喜霞, 等.均勻經(jīng)驗(yàn)過程精確漸近性的一個(gè)注記 [J].吉林大學(xué)學(xué)報(bào): 理學(xué)版, 2011, 49(5): 844-848.(WU Suwen, ZHANG Yong, LI Xixia, et al.A Note on the Precise Asymptotics for Uniform Empirical Processes [J].Journal of Jilin University: Science Edition, 2011, 49(5): 844-848.)

[13]李云霞.長(zhǎng)程相依過程關(guān)于矩完全收斂的精確漸近性質(zhì) [J].高校應(yīng)用數(shù)學(xué)學(xué)報(bào): A輯, 2013, 28(1): 23-33.(LI Yunxia.Precise Asymptotics for Complete Moment Convergence of Long Memory Processes [J].Appl Math J Chinese Univ: Ser A, 2013, 28(1): 23-33.)

[14]Gut A, Steinebach J.Precise Asymptotics: A General Approach [J].Acta Math Hungar, 2013, 138(4): 365-385.

[15]WANG Qiying, LIN Yanxia, Gulati C M.Asymptotics for Moving Average Processes with Dependent Innovations [J].Statist Probab Lett, 2001, 54(4): 347-356.

[16]WANG Qiying, LIN Yanxia, Gulati C M.Strong Approximation for Long Memory Processes with Applications [J].J Theoret Probab, 2003, 16(2): 377-389.

[17]WANG Lihong.Change-in-Mean Problem for Long Memory Time Series Models with Applications [J].J Stat Comput Simul, 2008, 78(7/8): 653-668.

[18]WU Weibiao, HUANG Yinxiao, ZHENG Wei.Covariances Estimation for Long-Memory Processes [J].Adv in Appl Probab, 2010, 42(1): 137-157.

[19]LI Yunxia, XU Jianjun, ZHANG Lixin.Testing for Changes in the Mean or Variance of Long Memory Processes [J].Acta Math Sin: Engl Ser, 2010, 26(12): 2443-2460.

[20]Palma W.On the Sample Mean of Locally Stationary Long-Memory Processes [J].J Statist Plann Inference, 2010, 140(12): 3764-3774.

[21]Beran J, Das B, Schell D.On Robust Tail Index Estimation for Linear Long-Memory Processes [J].J Time Series Anal, 2012, 33(3): 406-423.

[22]Khan M A R, Poskitt D S.Moment Tests for Window Length Selection in Singular Spectrum Analysis of Short-and Long-Memory Processes [J].J Time Series Anal, 2013, 34(2): 141-155.

[23]FU Ke’ang, Ng A C Y.A Note on the Strong Approximation for Long Memory Processes and Its Application [J].Statistics, 2013, 47(3): 511-520.

[24]Hall P, Heyde C C.Martingale Limit Theory and Its Application [M].New York: Academic Press, 1980.

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(guān)麗紅(1976—), 女, 滿族, 碩士, 講師, 從事概率統(tǒng)計(jì)與應(yīng)用數(shù)學(xué)的研究, E-mail: guanlihong14@163.com.

國(guó)家自然科學(xué)基金(批準(zhǔn)號(hào): 11371085)和吉林省教育廳“十二五”科技研究項(xiàng)目(批準(zhǔn)號(hào): 吉教科合字[2014]第526號(hào)).

O211.4

A

1671-5489(2014)06-1191-05

10.13413/j.cnki.jdxblxb.2014.06.16

趙立芹)