林君潔，鄧新，鮑瀟涵，王學軍

(安徽大學數(shù)學科學學院，安徽 合肥230039)

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非負WOD隨機變量的第k小矩不等式

林君潔，鄧新，鮑瀟涵，王學軍

(安徽大學數(shù)學科學學院，安徽 合肥230039)

WOD隨機變量; 矩不等式; 指數(shù)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)

則稱{Xn,n≥1}是WUOD隨機變量; 如果存在一個有限的實數(shù)序列{gL(n),n≥1}, 使得對任意的n≥1及所有xi∈(-∞,+∞),1≤i≤n, 滿足

(0.4)

1 相關(guān)結(jié)論

(1.1)

設x為一正數(shù),Gamma函數(shù)定義為

(1.2)

(1.3)

如果進一步假定ξ1,ξ2,…,ξn是獨立的, 則

(1.4)

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

(1.5)

(1.6)

2 主要結(jié)果

(2.1)

其中g(shù)(n)=max(gU(n),gL(n)). 定理2.1的證明 記

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的證明 不失一般性, 假定xi>0,i=1,2,…,n. 記

(2.3)

由定理2.1知

(2.4)

因此, 由(2.3)式和(2.4)式可得

(2.5)

特別地, 若{fn,n≥1}和{ξn,n≥1} 均為標準正態(tài)隨機變量序列, 則對任意的n≥1, 有

(2.6)

推論2.1的證明 由定理A知,

再由定理2.2和上面不等式可得,

定理2.3 設β>0,p>0,{xn,n≥1}為一非降的正數(shù)序列, {ξn,n≥1}為一個滿足(0.2)式的非負WOD隨機變量序列. 則對任意的n≥2和2≤k≤n, 有

(2.7)

因此, 由上述不等式及引理1得

定理得證.

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(責任編輯 趙燕)

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

安徽省自然科學基金(1508085J06)；安徽高校優(yōu)秀拔尖人才培育資助項目(gxb-jZD2016005)；大學生創(chuàng)新創(chuàng)業(yè)訓練計劃項目(201610357001)資助

林君潔(1997-)，女，本科生；王學軍，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