對(duì)數(shù)正態(tài)分布最大值的逐點(diǎn)收斂速度

黃建文,庹中友,羊 毫

(遵義師范學(xué)院數(shù)學(xué)與計(jì)算科學(xué)學(xué)院,貴州遵義563002)

對(duì)數(shù)正態(tài)分布最大值的逐點(diǎn)收斂速度

黃建文,庹中友,羊 毫

(遵義師范學(xué)院數(shù)學(xué)與計(jì)算科學(xué)學(xué)院,貴州遵義563002)

對(duì)數(shù)正態(tài)分布是統(tǒng)計(jì)、生物和其他一些領(lǐng)域應(yīng)用最廣泛的分布之一，并且在理論分析和廣泛的應(yīng)用中將正態(tài)分布轉(zhuǎn)到對(duì)數(shù)正態(tài)分布是流行的.作者研究了對(duì)數(shù)正態(tài)分布的收斂速度，得到逐點(diǎn)收斂速度.

對(duì)數(shù)正態(tài)分布；最大值；收斂速度

極值理論中某些分布的收斂速度是一個(gè)重要的問(wèn)題，同時(shí)也是一個(gè)難點(diǎn)問(wèn)題。近些年來(lái)，越來(lái)越多的學(xué)者投入到這一問(wèn)題的研究中來(lái)，而且取得了豐碩的成果[1-5].

對(duì)數(shù)正態(tài)分布的概率密度函數(shù)定義:對(duì)于x＞0,

廖昕和彭作祥[6]在2012年給出對(duì)數(shù)正態(tài)分布的尾部1-F(x)與其密度函數(shù)f(x)之間的關(guān)系式：當(dāng)x→∞時(shí)，

H←(y)=inf{x∶H(x)＞y}表示函數(shù)H(x)的左連續(xù)逆.

為了得到文章的主要結(jié)論,下面給出幾個(gè)引理.

1 輔助引理

廖昕和彭作祥[6]給出同服從對(duì)數(shù)正態(tài)分布獨(dú)立隨機(jī)變量序列的最大值的極限分布的定理,在這里引用來(lái)作為一個(gè)引理并且給出另外一種證明方法.

3 主要結(jié)果

[1]Hall P.On the Rate of Convergence of Normal Extremes[J]. Journal Applied Probability,1979,16(2):433-439.

[2]Xiong Fang,PengZuo-xiang.The RateofConvergenceofExtremes for the Erlang Distribution[J].西南大學(xué)學(xué)報(bào)(自然科學(xué)版),2011,33(3):5-8.

[3]Lin Fu-ming,Zhang Xin-hua,Peng Zuo-xiang,et al.On the rate of convergence of STSD extremes[J].Communications in Statistics-Theory and Methods,2011,40(10):1795-1806.

[4]Chen Shou-quan,Huang Jian-wen.Rates of Convergence of Extreme for Asymmetric Normal Distribution[J].Statistics and Probability Letters,2014,(84):158-168.

[5]劉豹,付穎.麥克斯韋分布的逐點(diǎn)收斂速度[J].西南大學(xué)學(xué)報(bào)(自然科學(xué)版),2013,35(5):80-83.

[6]Liao X,Peng Z.Convergence rates of limit distribution of maxima of lognormal samples[J].Journal of Mathematical Analysis and Applications,2012,395:643-653.

[7]Leadbetter M R,Lindgren G,Rootzen H.Extremes and Re lated Properties of Random Sequences and Processes[M].New York:Springer,1983.

[8]Resnick S I.Extreme value,Regular Variation,and Point Pro cesses[M].New York:Springer-Verlag,1987.

（責(zé)任編輯：朱 彬）

The Point-wise Rate of Convergence of Maxima for Logarithm Normal Distribution

HUANG Jian-wen,TUO Zhong-you,YANG Hao
(School of Mathematics and Computational Science,Zunyi Normal College,Zunyi 563002,China)

The lognormal distribution is one of the most widely applied distributions in statistics,biology andsome other disciplines.Furthermore,it is popular in theoretical analysis and wide applications that the normal distribution is carried over to logarithmic normal one. In this paper,we study convergence rate of the logarithm normal distribution and derive the point-wise convergence rate.

logarithm normal distribution;maxima;convergence rate

O211

A

1009-3583（2014）-0068-03

2013-12-23

黃建文，男，甘肅甘谷人，遵義師范學(xué)院數(shù)學(xué)與計(jì)算科學(xué)學(xué)院助教，碩士，主要從事概率論中極值理論和極限定理的研究。