非负WOD随机变量的第k小矩不等式

林君洁，邓新，鲍潇涵，王学军

(安徽大学数学科学学院，安徽 合肥230039)



非负WOD随机变量的第k小矩不等式

林君洁，邓新，鲍潇涵，王学军

(安徽大学数学科学学院，安徽 合肥230039)

WOD随机变量; 矩不等式; 指数不等式

0 引言

P(|ξ|≤t)≤αt, ∀t≥0

(0.1)

P(|ξ|>t)≤e-βt, ∀t≥0

(0.2)

(0.3)

则称{Xn,n≥1}是WUOD随机变量; 如果存在一个有限的实数序列{gL(n),n≥1}, 使得对任意的n≥1及所有xi∈(-∞,+∞),1≤i≤n, 满足

(0.4)

1 相关结论

(1.1)

设x为一正数,Gamma函数定义为

(1.2)

(1.3)

如果进一步假定ξ1,ξ2,…,ξn是独立的, 则

(1.4)

其中Γ(·)是Gamma函数. 定理B[2]设α>0,β>0,p>0,2≤k≤n. 设0

(1.5)

(1.6)

2 主要结果

(2.1)

其中g(n)=max(gU(n),gL(n)). 定理2.1的证明 记

Ai(t)={ω:xiξi(ω)>t}={ω:ξi(ω)>t/xi},i=1,2,…,n,

P(Ai(t))≤e-βt/xi,i=1,2,…,n.由(0.3)式及上述不等式得,

(2.2)

定理2.2的证明 不失一般性, 假定xi>0,i=1,2,…,n. 记

(2.3)

由定理2.1知

(2.4)

因此, 由(2.3)式和(2.4)式可得

(2.5)

特别地, 若{fn,n≥1}和{ξn,n≥1} 均为标准正态随机变量序列, 则对任意的n≥1, 有

(2.6)

推论2.1的证明 由定理A知,

再由定理2.2和上面不等式可得,

定理2.3 设β>0,p>0,{xn,n≥1}为一非降的正数序列, {ξn,n≥1}为一个满足(0.2)式的非负WOD随机变量序列. 则对任意的n≥2和2≤k≤n, 有

(2.7)

因此, 由上述不等式及引理1得

定理得证.

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(责任编辑 赵燕)

Moment inequalities of thek-minimum for nonnegative widely orthant dependent random variables

LIN Junjie，DENG Xin，BAO Xiaohan，WANG Xuejun

(School of Mathematical Sciences, Anhui University, Hefei 230039, China)

2017-02-26

安徽省自然科学基金(1508085J06)；安徽高校优秀拔尖人才培育资助项目(gxb-jZD2016005)；大学生创新创业训练计划项目(201610357001)资助

林君洁(1997-)，女，本科生；王学军，通信作者，教授，E-mail:wxjahdx 2000@126.com

1000-2375(2017)03-0248-05

O211.4

A

10.3969/j.issn.1000-2375.2017.03.007