西南大学学报 (自然科学版)  2019, Vol. 41 Issue (1): 72-77.  DOI: 10.13718/j.cnki.xdzk.2019.01.011
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  • 基于线性和条件下的失真风险测度尾部渐近性质    [PDF全文]
    邹沛清, 陈守全     
    西南大学 数学与统计学院, 重庆 400715
    摘要:对随机权重的次序统计量线性和条件下的失真风险测度尾部进行了讨论,并得到了相应的一些渐近性质.
    关键词极值理论    失真风险测度    失真函数    极值吸引场    正则变换    

    设{Xii≥1}为具有共同分布F的独立随机变量序列.若存在常数an>0,bn$\mathbb{R}$使得

    $ \mathop {\lim }\limits_{n \to \infty } P\left( {{a_n}\left( {\mathop {\max }\limits_{1 \leqslant i \leqslant n} {X_i} - {b_n}} \right) \leqslant x} \right) = G\left( x \right),x \in \mathbb{R} $

    其中G为非退化分布函数,则称F属于G的吸引场,记为FD(G).由文献[1]或者文献[2]可知G有3种类型.

    设(Ω$\mathscr{F} $$\mathscr{P}$)为概率空间,记L+($\mathscr{P}$)为其中非负随机变量分布集合.定义Hg(X)为随机变量X在失真函数g下的失真风险测度(文献[3]关于Hg(X)和g(x)给出了更为具体的定义),

    $ {H_g}\left( X \right) = \int_0^\infty {g\left( {1 - {F_X}\left( x \right)} \right){\rm{d}}x} = \int_0^\infty {g\left( {{{\bar F}_X}\left( x \right)} \right){\rm{d}}x} $

    其中FXL+($\mathscr{P}$)为随机变量X的分布函数.

    Xii=1,2,…,n为非负随机变量序列,Xini=1,2,…,n为对应的次序统计量,则X1:nX2:n≤…≤Xnn.本文目的为讨论当$\mathop {\lim }\limits_{t \to \infty } {\mkern 1mu} l\left( t \right) = {x_0}$(其中x0=sup{xF(x)<1}为上端点)时,

    $ {H_g}\left( {\theta {X_k}\left| {L\left( \mathit{\boldsymbol{C}} \right) > l\left( t \right)} \right.} \right) = \int_0^\infty {g\left( {P\left( {\theta {X_k} > w\left| {L\left( \mathit{\boldsymbol{C}} \right) > l\left( t \right)} \right.} \right)} \right){\rm{d}}w} ,1 \le k \le n,0 < \theta < 1 $

    的渐近性质,其中:L(C):=$\sum\limits_{i=1}^{n}{{{C}_{i}}{{X}_{n-i+1:n}}} $为次序统计量的线性和,C =(C1C2,…,Cn)为常数向量,0<θ≤1. Ci为随机权重并且C1>0.记θ=Ci,如果Xk=Xn-i+1:n.文献[4]给出了权重和的渐近性质.文献[5]得到了一致风险测度下的渐近性质.文献[6]给出了研究求和尾部风险的渐近性质的方法.文献[3]提供了在正则变换下失真风险测度的尾部性质.

    1 预备知识

    X1,…,Xn为非负随机变量,其对应的分布函数为F1,…,Fn并且满足:

    $ \mathop {\lim }\limits_{t \to \infty } \frac{{P\left( {{X_i} > l\left( t \right)} \right)}}{{P\left( {{X_1} > l\left( t \right)} \right)}} = {\lambda _i} \in \left[ {0,\infty } \right)\;\;\;\;\;\;\;i = 1,2, \cdots ,n $ (1)

    其中当F1D(Φα)或者F1D(Λ)时,l(t)=t;当F1D(Ψα)时,l(t)=x0-t-1.

    定义lower Matuszewska指标为

    $ \alpha _F^ * = {\rm{sup}}\left\{ { - \frac{{\log {{\bar F}^ * }\left( x \right)}}{{\log x}};x > 1} \right\} \in \left[ {0,\infty } \right] $

    其中

    $ {{\bar F}^ * }\left( x \right) = \mathop {\lim \sup }\limits_{t \to \infty } \frac{{\bar F\left( {l\left( {tx} \right)} \right)}}{{\bar F\left( {l\left( t \right)} \right)}} $

    条件1  1)当s→0时,失真函数g满足g(s)=O(sβ),记Ωg={β>0;g(s)=O(sβ),当s∈0}≠ø.

    2) 设XYL+($\mathscr{P}$),对应分布函数为FG,当t→∞时,满足F(l(t))=O(G(l(t))),极限

    $ \mathop {\lim }\limits_{t \to \infty } P\left( {X > l\left( {tx} \right)\left| {Y > l\left( t \right)} \right.} \right) = h\left( x \right) \in \left[ {0,1} \right] $

    几乎处处存在,其中x>0.

    对任意a≥0,记Δa={xahx处存在},则Δ0h的定义域.显然,(a,∞)\Δa的勒贝格测度为0,因此Δa在(a,∞)中稠密. h(x)在Δ0上非增.

    引理1  设条件2中2)成立.

    1) 如果0<αF*≤∞,那么$\mathop {\lim }\limits_{x \to \infty ,x \in {\mathit{\Delta } _0}} {\mkern 1mu} h\left( t \right) = 0$.

    2) 当1∈Δ0时,记

    $ h\left( 1 \right) = \mathop {\lim }\limits_{t \to \infty } \frac{{\bar F\left( {l\left( t \right)} \right)}}{{\bar G\left( {l\left( t \right)} \right)}} > 0 $

    若存在x1Δ1满足h(x1)>0,则当0≤α<∞时,有F(l(t))∈RV-α,进一步对所有x≥1,h(x)=h(1)x-α成立.

    定理1  设条件1成立.如果$ \frac{1}{{{\beta }^{*}}}$αF*≤∞,其中β*=sup{Ωg},记l(t)=x0-t-1,则

    $ \mathop {\lim }\limits_{t \to \infty } t{H_g}\left( {X\left| {Y > {x_0} - {t^{ - 1}}} \right.} \right) = \int_0^\infty {\frac{1}{{{x^2}}}g\left( {h\left( x \right)} \right){\rm{d}}x} $

    引理1和定理1的证明与文献[7]中的引理3.1和定理3.1的证明类似.

    现在开始讨论当t→∞时,Hg(θXk|L(C)>l(t))基于下面几个假定条件的渐近性.

    条件2  设F1D(Φα),并且

    $ \mathop {\lim }\limits_{t \to \infty } \frac{{P\left( {{X_i} > t,{X_j} > t} \right)}}{{P\left( {{X_1} > t} \right)}} = 0 $
    $ \mathop {\lim }\limits_{t \to \infty } \frac{{P\left( {\tilde C{X_i} > t,\tilde C{X_j} > t} \right)}}{{P\left( {{X_1} > t} \right)}} = 0 $

    其中$\widetilde C = \mathop {\max }\limits_{2 \le i \le n} {\mkern 1mu} \left\{ {{C_i}} \right\}$.

    条件3  设F1D(Ψα),并且

    $ \mathop {\lim }\limits_{t \to \infty } \mathop {\max }\limits_{1 \le i \ne j \le n} \frac{{P\left( {{C^ * }{X_i} > {x_0} - {{\left( {tx} \right)}^{ - 1}},{C^ * }{X_j} > {x_0} - {{\left( {tx} \right)}^{ - 1}}} \right)}}{{P\left( {{C_1}{X_1} > {x_0} - {t^{ - 1}}} \right)}} = 0,x > 0 $

    其中C*:= $\mathop {\max }\limits_{1 \le i \le n} {\mkern 1mu} {C_i}$.

    条件4  设F1D(Λ),并且

    $ \mathop {\lim }\limits_{t \to \infty } \mathop {\max }\limits_{1 \le i \ne j \le n} \frac{{P\left( {{C^ * }{X_i} > a\left( t \right)x,{C^ * }{X_j} > a\left( t \right)x} \right)}}{{P\left( {{C_1}{X_1} > t} \right)}} = 0,x > 0 $
    $ \mathop {\lim }\limits_{t \to \infty } \mathop {\max }\limits_{1 \le i \ne j \le n} \frac{{P\left( {{C^ * }{X_i} > {L_{ij}}a\left( t \right),{C^ * }{X_j} > {L_{ij}}a\left( t \right)} \right)}}{{P\left( {{C_1}{X_1} > t} \right)}} = 0,{L_{ij}} > 0 $

    因此由文献[4]中的定理3.1以及文献[8]中的定理3.1可知,当t→∞时,有

    $ P\left( {L\left( \mathit{\boldsymbol{C}} \right) > l\left( t \right)} \right) \sim P\left( t \right)\left( {{C_1}{X_{n:n}} > l\left( t \right)} \right) \sim \sum\limits_{i = 1}^n {{\lambda _i}P\left( {{C_1}{X_1} > l\left( t \right)} \right)} \sim \sum\limits_{i = 1}^n {P\left( {{C_1}{X_i} > l\left( t \right)} \right)} $ (2)

    引理2  1)设条件2和公式(1)成立,则

    $ \mathop {\lim }\limits_{t \to \infty } P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right) = {\lambda _k}{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)^{ - 1}}{I_{\left\{ {0 < x \le \frac{\theta }{{{C_1}}}} \right\}}} + {\lambda _k}{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)^{ - 1}}{\left( {\frac{\theta }{{{C_1}}}} \right)^\alpha }{x^{ - \alpha }}{I_{\left\{ {x > \frac{\theta }{{{C_1}}}} \right\}}} $

    2) 设条件3和公式(1)成立,则

    $ \mathop {\lim }\limits_{t \to \infty } P\left( {\theta {X_k} > {x_0} - {{\left( {tx} \right)}^{ - 1}}\left| {L\left( \mathit{\boldsymbol{C}} \right) > {x_0} - {{\left( t \right)}^{ - 1}}} \right.} \right) = {\lambda _k}{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)^{ - 1}}{I_{\left\{ {0 < x \le \frac{\theta }{{{C_1}}}} \right\}}} + 0 \cdot {I_{\left\{ {x > \frac{\theta }{{{C_1}}}} \right\}}} $

    3) 设条件4和公式(1)成立,则

    $ \mathop {\lim }\limits_{t \to \infty } P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right) = {\lambda _k}{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)^{ - 1}}{I_{\left\{ {0 < x \le \frac{\theta }{{{C_1}}}} \right\}}} + 0 \cdot {I_{\left\{ {x > \frac{\theta }{{{C_1}}}} \right\}}} $

      对于0<x$\frac{\theta }{{{C}_{1}}}$时,存在t0>0充分大,当tt0时,由公式(2)可得

    $ \begin{array}{l} P\left( {\theta {X_k} > l\left( {tx} \right),L\left( \mathit{\boldsymbol{C}} \right) > l\left( t \right)} \right) = P\left( {L\left( \mathit{\boldsymbol{C}} \right) > l\left( t \right)} \right) - P\left( {\theta {X_k} \le l\left( {tx} \right),L\left( \mathit{\boldsymbol{C}} \right) > l\left( t \right)} \right) \le \\ P\left( {L\left( \mathit{\boldsymbol{C}} \right) > l\left( t \right)} \right) - P\left( {\theta {X_k} \le l\left( {tx} \right),\bigcup\limits_{i = 1,i \ne k}^n {\left\{ {{C_1}{X_i} > l\left( t \right)} \right\}} } \right) = \\ P\left( {L\left( \mathit{\boldsymbol{C}} \right) > l\left( t \right)} \right) - P\left( {\bigcup\limits_{i = 1,i \ne k}^n {\left\{ {{C_1}{X_i} > l\left( t \right)} \right\}} } \right) + P\left( {\theta {X_k} > l\left( {tx} \right),\bigcup\limits_{i = 1,i \ne k}^n {\left\{ {{C_1}{X_i} > l\left( t \right)} \right\}} } \right) \le \\ P\left( {L\left( \mathit{\boldsymbol{C}} \right) > l\left( t \right)} \right) - \sum\limits_{i = 1,i \ne k}^n {P\left( {{C_1}{X_i} > l\left( t \right)} \right)} + \\ \sum\limits_{1 \le i < j \le n,i \ne k \ne j}^n {P\left( {{C_1}{X_i} > l\left( t \right),{C_1}{X_j} > l\left( t \right)} \right)} + \sum\limits_{i = 1,i \ne k}^n {P\left( {\theta {X_k} > l\left( t \right),{C_1}{X_i} > l\left( t \right)} \right)} \end{array} $ (3)

    首先证明1).如果条件2和公式(1)成立,则对于l(t)=tt→∞时由(3)式可得

    $ \mathop {\lim \sup }\limits_{t \to \infty } \frac{{P\left( {\theta {X_k} > tx,L\left( \mathit{\boldsymbol{C}} \right) > t} \right)}}{{P\left( {{X_1} > t} \right)}} \le C_1^\alpha \left( {\sum\limits_{i = 1}^n {{\lambda _i}} - \sum\limits_{i = 1,i \ne k}^n {{\lambda _i}} } \right) + 0 + 0 = C_1^\alpha {\lambda _k} $ (4)

    显然,

    $ \mathop {\lim \inf }\limits_{t \to \infty } \frac{{P\left( {\theta {X_k} > tx,L\left( \mathit{\boldsymbol{C}} \right) > t} \right)}}{{P\left( {{X_1} > t} \right)}} \ge \mathop {\lim \inf }\limits_{t \to \infty } \frac{{P\left( {{C_1}{X_k} > t} \right)}}{{P\left( {{X_1} > t} \right)}} = C_1^\alpha {\lambda _k} $ (5)

    因此,联合式(1),(4)和(5),有

    $ \mathop {\lim }\limits_{t \to \infty } P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right) = {\lambda _k}{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)^{ - 1}} $

    另一方面,对于x$\frac{\theta }{{{C}_{1}}}$,因为F1(t)∈D(Φα)有

    $ \begin{array}{l} \mathop {\lim }\limits_{t \to \infty } P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right) = \\ \mathop {\lim }\limits_{t \to \infty } \frac{{P\left( {\theta {X_k} > tx} \right)}}{{P\left( {L\left( \mathit{\boldsymbol{C}} \right) > t} \right)}} = \\ \mathop {\lim }\limits_{t \to \infty } \frac{{P\left( {\theta {X_k} > tx} \right)}}{{P\left( {\theta {X_1} > tx} \right)}}\frac{{P\left( {{C_1}{X_1} > t} \right)}}{{P\left( {L\left( \mathit{\boldsymbol{C}} \right) > t} \right)}}\frac{{P\left( {\theta {X_1} > tx} \right)}}{{P\left( {{C_1}{X_1} > t} \right)}} = \\ {\lambda _k}{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)^{ - 1}}{\left( {\frac{\theta }{{{C_1}}}} \right)^\alpha }{x^{ - \alpha }} \end{array} $

    因此结论成立. 2)和3)的证明和1)类似.

    2 主要结论及其证明

    定理2  1)设条件2和公式(1)成立.如果g(·)连续,且当s→0时,g(s)=O(sβ),则当t→∞时,

    $ \begin{array}{l} \mathop {\lim }\limits_{t \to \infty } \left( {P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right)} \right) = g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}} \right){I_{\left\{ {0 < x \le \frac{\theta }{{{C_1}}}} \right\}}} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}{{\left( {\frac{\theta }{{{C_1}}}} \right)}^\alpha }{x^{ - \alpha }}} \right){I_{\left\{ {x > \frac{\theta }{{{C_1}}}} \right\}}} \end{array} $

    关于x∈(0,∞)一致收敛.

    2) 设条件3和公式(1)成立.如果g(·)连续,且当s→0时,g(s)=O(sβ),则当t→∞时,

    $ \mathop {\lim }\limits_{t \to \infty } g\left( {P\left( {\theta {X_k} > {x_0} - {{\left( {tx} \right)}^{ - 1}}\left| {L\left( \mathit{\boldsymbol{C}} \right) > {x_0} - {{\left( {tx} \right)}^{ - 1}}} \right.} \right)} \right) = g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}} \right) $

    关于x$\left( 0, \frac{\theta }{{{C}_{1}}} \right]$一致收敛.

    3) 设条件4和公式(1)成立.如果g(·)连续,并且当s→0时,g(s)=O(sβ),则当t→∞时,

    $ \mathop {\lim }\limits_{t \to \infty } g\left( {P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right)} \right) = g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}} \right) $

    关于x$\left( 0, \frac{\theta }{{{C}_{1}}} \right]$一致收敛.

      先证明1).因为g(·)在[0, 1]上连续,所以g(·)在[0, 1]上一致连续.因此对任意的ε>0,存在δ>0,使得对所有xx*∈[0, 1],当|x-x*|<δ时,有|g(x)-g(x*)|<ε.并且由引理2的一致收敛性可知,可选择δ>0,存在N>0,当tN时:

    $ \left| {P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right) - {\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}} \right| < \frac{\delta }{{\rm{2}}},x \in \left( {0,\frac{\theta }{{{C_1}}}} \right] $
    $ \left| {P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right) - {\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}{{\left( {\frac{\theta }{{{C_1}}}} \right)}^\alpha }{x^{ - \alpha }}} \right| < \frac{\delta }{{\rm{2}}},x \in \left( {\frac{\theta }{{{C_1}}},\infty } \right) $

    因为P(θXktx|L(C)>t)∈[0, 1]且λk($\sum\limits_{i=1}^{n}{{{\lambda }_{i}}}$)-1∈[0, 1],当tN时,

    $ \begin{array}{l} \left| {g\left( {P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right)} \right) - g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}} \right)} \right|{I_{\left\{ {0 < x \le \frac{\theta }{{{C_1}}}} \right\}}} + \\ \left| {g\left( {P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right)} \right) - g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}{{\left( {\frac{\theta }{{{C_1}}}} \right)}^\alpha }{x^{ - \alpha }}} \right)} \right|{I_{\left\{ {x > \frac{\theta }{{{C_1}}}} \right\}}} < \varepsilon \end{array} $

    因此一致性成立. 2)和3)的证明和1)相似,证明省略.

    定理3  1)设条件2和公式(1)成立.如果g(·)为任意一失真函数,并且当s→0时,满足g(s)=O(sβ),则当t→∞时,

    $ \begin{array}{l} \mathop {\lim }\limits_{t \to \infty } \int_0^\infty {g\left( {P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right)} \right){\rm{d}}x} = \\ \int_0^{\frac{\theta }{{{C_1}}}} {g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}} \right){\rm{d}}x} + \int_{\frac{\theta }{{{C_1}}}}^\infty {g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}{{\left( {\frac{\theta }{{{C_1}}}} \right)}^\alpha }{x^{ - \alpha }}} \right){\rm{d}}x} \end{array} $

    2) 设条件3和公式(1)成立.如果g(·)为任意一失真函数,并且当s→0时,满足g(s)=O(sβ),则对于0<x$\frac{\theta }{{{C}_{1}}}$,当t→∞时,

    $ \mathop {\lim }\limits_{t \to \infty } \int_0^{\frac{\theta }{{{C_1}}}} {g\left( {P\left( {\theta {X_k} > l\left( {tx} \right)\left| {L\left( \mathit{\boldsymbol{C}} \right) > l\left( t \right)} \right.} \right)} \right){\rm{d}}x} = \int_0^{\frac{\theta }{{{C_1}}}} {g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}} \right){\rm{d}}x} $

    3) 设条件4和公式(1)成立.如果g(·)为任意一失真函数,并且当s→0时,满足g(s)=O(sβ),则对于0<x$\frac{\theta }{{{C}_{1}}}$,当t→∞时,

    $ \mathop {\lim }\limits_{t \to \infty } \int_0^{\frac{\theta }{{{C_1}}}} {g\left( {P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right)} \right){\rm{d}}x} = \int_0^{\frac{\theta }{{{C_1}}}} {g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}} \right){\rm{d}}x} $

      先证明1).显然地,

    $ \int_0^{\frac{\theta }{{{C_1}}}} {g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}} \right){\rm{d}}x} + \int_{\frac{\theta }{{{C_1}}}}^\infty {g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}{{\left( {\frac{\theta }{{{C_1}}}} \right)}^\alpha }{x^{ - \alpha }}} \right){\rm{d}}x} < \infty $

    因为g(·)为非增函数且有界,则g(·)的不连续点的集合是至多可数的且其勒贝格测度为零.也就是说

    $ \begin{array}{l} \mathop {\lim }\limits_{t \to \infty } g\left( {P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right)} \right) = g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}} \right){I_{\left\{ {0 < x \le \frac{\theta }{{{C_1}}}} \right\}}} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}{{\left( {\frac{\theta }{{{C_1}}}} \right)}^\alpha }{x^{ - \alpha }}} \right){I_{\left\{ {x > \frac{\theta }{{{C_1}}}} \right\}}} \end{array} $ (6)

    x∈(0,∞)上几乎处处收敛.

    由第三littlewood原理,存在集合A⊆(0,∞),使得μ(A)= $\int_{A}{\text{d}\mathit{x}}\le \frac{\varepsilon }{4}$,其中μ(·)为勒贝格测度.又因为公式(6)在(0,∞)\A上一致收敛,即存在一个N>0,使得当tN时:

    $ \left| {g\left( {P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right)} \right) - g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}} \right)} \right| < \frac{{\varepsilon {C_1}}}{{4\theta }},x \in \left( {0,\frac{\theta }{{{C_1}}}} \right]\backslash A $
    $ \left| {g\left( {P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right)} \right) - g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}{{\left( {\frac{\theta }{{{C_1}}}} \right)}^\alpha }{x^{ - \alpha }}} \right)} \right| < \frac{\varepsilon }{M},x \in \left( {\frac{\theta }{{{C_1}}},\infty } \right)\backslash A $

    其中M充分大使得

    $ \int_{\frac{\theta }{{{C_1}}}}^\infty {\frac{\varepsilon }{M} \le \frac{\varepsilon }{4}} $

    因为g(·)有界且小于1,所以当tN时,

    $ \begin{array}{l} \left| {\int_0^\infty {g\left( {P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right)} \right){\rm{d}}x} - \int_0^{\frac{\theta }{{{C_1}}}} {g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}} \right){\rm{d}}x} - } \right.\\ \left. {\int_{\frac{\theta }{{{C_1}}}}^\infty {g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}{{\left( {\frac{\theta }{{{C_1}}}} \right)}^\alpha }{x^{ - \alpha }}} \right){\rm{d}}x} } \right| \le \\ \int_{A \cap \left( {0,\frac{\theta }{{{C_1}}}} \right]} {\left| {g\left( {P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right)} \right) - g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}} \right)} \right|{\rm{d}}x} + \\ \int_{A \cap \left( {\frac{\theta }{{{C_1}}},\infty } \right)} {\left| {g\left( {P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right)} \right) - g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}{{\left( {\frac{\theta }{{{C_1}}}} \right)}^\alpha }{x^{ - \alpha }}} \right)} \right|{\rm{d}}x} + \\ \int_{\left( {0,\frac{\theta }{{{C_1}}}} \right]\backslash A} {\left| {g\left( {P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right)} \right) - g\left( {{\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}} \right)} \right|{\rm{d}}x} + \\ \int_{\left( {\frac{\theta }{{{C_1}}},\infty } \right)\backslash A} {\left| {g\left( {P\left( {\theta {X_k} > tx\left| {L\left( \mathit{\boldsymbol{C}} \right) > t} \right.} \right)} \right) - g\left( {C_1^\alpha {\lambda _k}{{\left( {\sum\limits_{i = 1}^n {{\lambda _i}} } \right)}^{ - 1}}{{\left( {\frac{\theta }{{{C_1}}}} \right)}^\alpha }{x^{ - \alpha }}} \right)} \right|{\rm{d}}x} \le \\ 2\int_A {{\rm{d}}x} + \frac{\varepsilon }{4} + \frac{\varepsilon }{4} = \varepsilon \end{array} $

    因此结论得证. 2)和3)的证明和1)相似,省略.

    参考文献
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    Asymptotic Tail Probabilities of Distortion Risk Measurement Based on Linear Combinations of Order Statistics
    ZOU Pei-qing, CHEN Shou-quan     
    School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
    Abstract: In this note, we investigate the tail distortion risk for linear combinations of randomly weighted order statistics, and obtain the asymptotic properties.
    Key words: extreme value theory    distortion risk measurement    distortion function    max-domain of attraction    regular variation    
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