西南大学学报 (自然科学版)  2019, Vol. 41 Issue (1): 78-83.  DOI: 10.13718/j.cnki.xdzk.2019.01.012
0
Article Options
  • PDF
  • Abstract
  • Figures
  • References
  • 扩展功能
    Email Alert
    RSS
    本文作者相关文章
    彭茜
    周玮
    彭作祥
    欢迎关注西南大学期刊社
     

  • 幂赋范极值分布的条件矩刻画    [PDF全文]
    彭茜, 周玮, 彭作祥     
    西南大学 数学与统计学院, 重庆 400715
    摘要:{Xn}为独立同分布的离散型随机变量序列,其分布函数为Fx).得到了Fx)属于幂赋范极值分布吸引场的条件矩刻画.
    关键词幂赋范极值分布    条件矩    吸引场    

    设{Xn}是独立同分布随机变量序列,公共分布函数F(x),记其部分最大值${M_n} = \mathop {\max }\limits_{1 \le k \le n} \left\{ {{X_k}} \right\}$,若存在规范化常数序列αn>0,βn>0,使得

    $ \mathop {\lim }\limits_{n \to \infty } \mathbb{P}\left( {{M_x} \leqslant {\alpha _n}{{\left| x \right|}^{{\beta _n}}}{\text{sign}}\left( x \right)} \right) = \mathop {\lim }\limits_{n \to \infty } {F^n}\left( {{\alpha _n}{{\left| x \right|}^{{\beta _n}}}{\text{sign}}\left( x \right)} \right) = H\left( x \right) $ (1)

    其中sign(x)是符号函数,H(x)为非退化分布函数,且如果(1)式成立,则称F(x)属于幂赋范极值分布H(x)的吸引场,记作FDp(H).由文献[1]可知,H(x)是以下6种类型之一:

    $ {\rm{I}}:{H_{1,\alpha }}\left( x \right) = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} 0\\ \exp \left( { - {{\left( {\log x} \right)}^{ - \alpha }}} \right) \end{array}&\begin{array}{l} x \le 1\\ x > 1 \end{array} \end{array}} \right. $
    $ {\rm{II}}:{H_{2,\alpha }}\left( x \right) = \left\{ \begin{array}{l} 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \le 0\\ \exp \left( { - {{\left( { - \log x} \right)}^\alpha }} \right)\;\;\;\;\;\;\;0 < x < 1\\ 1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \ge 1 \end{array} \right. $
    $ {\rm{III}}:{H_{3,\alpha }}\left( x \right) = \left\{ \begin{array}{l} 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \le - 1\\ \exp \left( { - {{\left( { - \log \left( { - x} \right)} \right)}^{ - \alpha }}} \right)\;\;\;\;\;\;\; - 1 < x < 0\\ 1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \ge 0 \end{array} \right. $
    $ {\rm{IV}}:{H_{4,\alpha }}\left( x \right) = \left\{ \begin{array}{l} \exp \left( { - {{\left( {\log \left( { - x} \right)} \right)}^\alpha }} \right)\;\;\;\;\;\;x \le - 1\\ 1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \ge - 1 \end{array} \right. $
    $ {\rm{V}}:\mathit{\Phi }\left( x \right) = \left\{ \begin{array}{l} 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \le 0\\ \exp \left( { - \frac{1}{x}} \right)\;\;\;\;\;\;\;\;x > 0 \end{array} \right. $
    $ {\rm{VI}}:\mathit{\Psi }\left( x \right) = \left\{ \begin{array}{l} \exp \left( x \right)\;\;\;\;\;x < 0\\ 1\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \ge 0 \end{array} \right. $

    其中α为大于0的常数.

    文献[2]研究了6种类型的分布函数属于幂赋范条件下的吸引场的充要条件;文献[3]研究了在幂赋范下极值的矩收敛和密度函数收敛.使用条件矩刻画F属于给定的极值分布,在线性赋范情形文献[4]得到F属于Λ的极值吸引场的条件矩刻画;文献[5]得到F属于ΦαΨα的条件矩刻画.本文将研究在幂赋范条件下,F分别属于上述H1,αH2,αH3,αH4,αΨ(x)和Φ(x) 6种吸引场的条件矩刻画(见定理1-定理6).为证明主要结论,需要引用下面的结论.

    引理1[4]  设U${\mathbb{R} ^ + } \to {\mathbb{R} ^ + }$,且

    $ U\left( t \right) = \int_t^\infty {u\left( s \right){\rm{d}}s} $

    是有限的.如果

    $ \mathop {\lim }\limits_{t \to \infty } \frac{{tu\left( t \right)}}{{U\left( t \right)}} = \rho $ (2)

    那么U(t)∈RV-ρ.相反地,如果U(t)∈RV-ρρ>0,且u(t)>0为减函数,那么(2)式成立,且u(t)∈RV-ρ-1.

    为统一符号,定义r(F)=sup {x|F(x)<1}为分布函数F(x)的上端点.

    定理1  若rF=∞.定义

    $ {\mu _p}\left( t \right) = E\left\{ {{{\left( {\left( {\log \left| X \right|} \right) - t} \right)}^p}\left| {\left( {\log \left| X \right|} \right) > t} \right.} \right\} $

    如果FDp(H1,α),α>2,那么对于0≤pα-2,μp+2(t)<∞且t→∞时,

    $ \frac{{{\mu _p}\left( t \right){\mu _{p + 2}}\left( t \right)}}{{{{\left\{ {{\mu _{p + 1}}\left( t \right)} \right\}}^2}}} \to \frac{{\left( {p + 2} \right)\left( {\alpha - p - 1} \right)}}{{\left( {p + 1} \right)\left( {\alpha - p - 2} \right)}} $ (3)

    反之,当α>2时,对0≤pα-2,如果(3)式成立,那么FDp(H1,α).

      关于必要性:由文献[2]之定理2.1知,FDp(H1,α)的充要条件为对于y>0时,

    $ \mathop {\lim }\limits_{t \to \infty } \frac{{1 - F\left( {\exp \left( {ty} \right)} \right)}}{{1 - F\left( {\exp \left( t \right)} \right)}} = {y^{ - \alpha }} $

    α>2时有

    $ \begin{array}{l} {\mu _p}\left( t \right) = E\left\{ {{{\left( {\left( {\log \left| X \right|} \right) - t} \right)}^p}\left| {\left( {\log \left| X \right|} \right) > t} \right.} \right\} = \\ \;\;\;\;\;\;\;\;\;\;\;\int_{\exp \left( t \right)}^\infty {\frac{{{{\left( {\left( {\log \left| x \right|} \right) - t} \right)}\;^p}{\rm{d}}F\left( x \right)}}{{1 - F\left( {\exp \left( t \right)} \right)}}} = p{t^p}\int_1^\infty {{{\left( {u - 1} \right)}^{p - 1}}\frac{{1 - F\left( {\exp \left( {tu} \right)} \right)}}{{1 - F\left( {\exp \left( t \right)} \right)}}{\rm{d}}u} \to \\ \;\;\;\;\;\;\;\;\;\;\;p{t^p}\int_1^\infty {{{\left( {u - 1} \right)}^{p - 1}}{u^{ - \alpha }}{\rm{d}}u} ,\alpha > 0 = \frac{{\mathit{\Gamma }\left( {\alpha - p} \right)\mathit{\Gamma }\left( {p + 1} \right)}}{{\mathit{\Gamma }\left( \alpha \right)}}{t^p} \end{array} $ (4)

    显然,这对p=0也成立.由(4)式知(3)式成立.

    对于充分性,若(3)式成立.定义

    $ {J_p}\left( t \right) = {\left\{ {\mathit{\Gamma }\left( {p + 1} \right)} \right\}^{ - 1}}\int_{\exp \left( t \right)}^\infty {{{\left( {\left( {\log \left| x \right|} \right) - t} \right)}^p}{\rm{d}}F\left( x \right)} $

    对任意的δ≥-1,有

    $ \begin{array}{l} \int_t^\infty {{{\left( {u - t} \right)}^\delta }{J_p}\left( u \right){\rm{d}}u} = {\left\{ {\mathit{\Gamma }\left( {p + 1} \right)} \right\}^{ - 1}}\int_t^\infty {\int_{\exp \left( u \right)}^\infty {{{\left( {u - t} \right)}^\delta }{{\left( {\left( {\log \left| x \right|} \right) - u} \right)}^p}{\rm{d}}F\left( x \right){\rm{d}}u} } = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left\{ {\mathit{\Gamma }\left( {p + 1} \right)} \right\}^{ - 1}}\int_{\exp \left( t \right)}^\infty {\int_t^{\log \left| x \right|} {{{\left( {u - t} \right)}^\delta }{{\left( {\left( {\log \left| x \right|} \right) - u} \right)}^p}{\rm{d}}u{\rm{d}}F\left( x \right)} } = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{\mathit{\Gamma }\left( {\delta + 1} \right)}}{{\mathit{\Gamma }\left( {p + \delta + 2} \right)}}\int_{\exp \left( t \right)}^\infty {{{\left( {\left( {\log \left| x \right|} \right) - t} \right)}^{p + \delta }}{\rm{d}}F\left( x \right)} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathit{\Gamma }\left( {\delta + 1} \right){J_{\delta + p + 1}}\left( t \right) \end{array} $ (5)

    特别地,当t→∞时,

    $ {J_{p + 1}}\left( t \right) = \int_t^\infty {{J_p}\left( u \right){\rm{d}}u} \downarrow 0 $ (6)

    利用以下关系式

    $ {\mu _p}\left( t \right) = \frac{{\mathit{\Gamma }\left( {p + 1} \right)}}{{1 - F\left( {\exp \left( t \right)} \right)}}{J_p}\left( t \right) $

    则式(3)可写为

    $ \frac{{J\left( t \right){J_{p + 2}}\left( t \right)}}{{{{\left\{ {{J_{p + 1}}\left( t \right)} \right\}}^2}}} \to \frac{{\alpha - p - 1}}{{\alpha - p - 2}} $ (7)

    其中0≤pα-2.

    如果式(7)对p=0成立,那么设${J_0}^*\left(t \right) = \frac{{J_1^2\left(t \right)}}{{{J_2}\left(t \right)}}$,则$\frac{{{J_0}\left(t \right)}}{{{J_0}^*\left(t \right)}} \to \frac{{\alpha - 1}}{{\alpha - 2}}$.只需证J0*(t)∈RV-α.由于

    $ \begin{array}{l} {\left( {{J_0}^ * \left( t \right)} \right)^\prime } = {\left( {\frac{{J_1^2\left( t \right)}}{{{J_2}\left( t \right)}}} \right)^\prime } = \frac{{{J_1}^3\left( t \right)}}{{{J_2}^2\left( t \right)}}\left( {1 - \frac{{2{J_0}\left( t \right){J_2}\left( t \right)}}{{{J_1}^2\left( t \right)}}} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - \frac{1}{t}{J_0}^ * \left( t \right)\frac{t}{{f\left( t \right)}}\left( {2\frac{{{J_0}\left( t \right)}}{{{J_0}^ * \left( t \right)}} - 1} \right) \end{array} $

    其中$f\left(t \right) = \frac{{{J_2}\left(t \right)}}{{{J_1}\left(t \right)}}$.故${J_0}^*\left(t \right) = {J_0}^*\left({{z_0}} \right)\exp \left({ - \int_{{z_0}}^t {\frac{{g\left(u \right)}}{u}{\rm{d}}u} } \right)$,其中$g\left(t \right) = \frac{t}{{f\left(t \right)}}\left({2\frac{{{J_0}\left(t \right)}}{{{J_0}^*\left(t \right)}} - 1} \right)$.注意到$f'\left(t \right) = - 1 + \frac{{{J_0}\left(t \right)}}{{{J_0}^*\left(t \right)}} \to \frac{1}{{\alpha - 2}}, g\left(t \right) \to \frac{\alpha }{{\alpha - 2}}\left({\alpha - 2} \right) = \alpha $.

    由文献[4]的Karamata公式可知,J0*(t)∈RV-α.假设式(7)对于整数0<pα-2成立,定义

    $ {J_p}^ * \left( t \right) = \frac{{{{\left( {{J_{p + 1}}\left( t \right)} \right)}^2}}}{{{J_{p + 2}}\left( t \right)}} $

    那么有

    $ \frac{{{J_p}\left( t \right)}}{{{J_p}^ * \left( t \right)}} \to \frac{{\alpha - p - 1}}{{\alpha - p - 2}} $

    p=0的方法相同,很容易得到Jp(t)∈RV-(α-p).由式(6)和引理1,得JP-1RV-(α-p)-1.将上述步骤重复p次,就得到1-F(exp(t))=J0(t)∈RV-α.

    假设对实数0<pα-2式子(7)成立.与p为整数时情况相同,有Jp(t)∈RV-(α-p).不失一般性,设0<p<1.在δ≥-1,αδ+p+1情况下,有

    $ {J_{\delta + p + 1}}\left( t \right) = \frac{1}{{\mathit{\Gamma }\left( {\delta + 1} \right)}}\int_t^\infty {{{\left( {u - t} \right)}^\delta }{J_p}\left( u \right){\rm{d}}u} = \frac{{{t^{\delta + 1}}}}{{\mathit{\Gamma }\left( {\delta + 1} \right)}}\int_t^\infty {{{\left( {y - 1} \right)}^\delta }{J_p}\left( {ty} \right){\rm{d}}y} $

    因此,

    $ \begin{array}{*{20}{c}} {\frac{{{J_{\delta + p + 1}}\left( {tx} \right)}}{{{t^{\delta + 1}}{J_p}\left( t \right)}} = \frac{{{x^{\delta + 1}}}}{{\mathit{\Gamma }\left( {\delta + 1} \right)}}\int_1^\infty {{{\left( {y - 1} \right)}^\delta }\frac{{{J_p}\left( {txy} \right)}}{{{J_p}\left( t \right)}}{\rm{d}}y} \to }\\ {\frac{{{x^{ - \alpha + p + 1 + \delta }}}}{{\mathit{\Gamma }\left( {\delta + 1} \right)}}\int_1^\infty {{{\left( {y - 1} \right)}^\delta }{y^{ - \alpha + p}}{\rm{d}}y} = }\\ {{x^{ - \alpha + p + 1 + \delta }}\frac{{\mathit{\Gamma }\left( {\alpha - p - \delta - 1} \right)}}{{\mathit{\Gamma }\left( {\alpha - p} \right)}}} \end{array} $

    因此,Jδ+p+1(t)∈RV-α+p+1+δ.如果δ=-p>-1,那么J1(t)∈RV-(α-1),可得

    $ 1 - F\left( {\exp \left( t \right)} \right) = {J_0}\left( t \right) \in R{V_{ - \alpha }} $

    证毕.

    定理2  若0<xF<∞,定义

    $ {\mu _p}\left( t \right) = E\left\{ {{{\left( {\left( { - \log \left| {\frac{X}{{r\left( F \right)}}} \right|} \right) - \frac{1}{t}} \right)}^p}\left( { - \log \left| {\frac{X}{{r\left( F \right)}}} \right|} \right) > \frac{1}{t}} \right\} $

    如果FDp(H2α),α>2,那么对于0≤pα-2,μp+2(t)<∞且t→∞时,

    $ \frac{{{\mu _p}\left( t \right){\mu _{p + 2}}\left( t \right)}}{{{{\left\{ {{\mu _{p + 1}}\left( t \right)} \right\}}^2}}} \to \frac{{\left( {p + 2} \right)\left( { - \alpha - p - 1} \right)}}{{\left( {p + 1} \right)\left( { - \alpha - p - 2} \right)}} $ (8)

    反之,当α>2时,对0≤pα-2,如果μp+2(t)<∞且式子(8)成立,那么FDp(H2α).

      必要性相的证明似于定理1中(4)的证明,利用文献[2]中定理2.2给出的FDp(H2α)充要条件为当y>0时,

    $ \mathop {\lim }\limits_{t \to \infty } \frac{{1 - F\left( {r\left( F \right)\exp \left( { - \frac{y}{t}} \right)} \right)}}{{1 - F\left( {r\left( F \right)\exp \left( { - \frac{1}{t}} \right)} \right)}} = {y^\alpha } $

    对充分性,定义

    $ {J_p}\left( t \right) = {\left\{ {\mathit{\Gamma }\left( {p + 1} \right)} \right\}^{ - 1}}\int_{r\left( F \right)\exp \left( { - \frac{1}{t}} \right)}^{r\left( F \right)} {{{\left( {\left( { - \log \left| {\frac{x}{{r\left( F \right)}}} \right|} \right) - \frac{1}{t}} \right)}^p}{\rm{d}}F\left( x \right)} $

    $ \int_t^\infty {{{\left( {u - t} \right)}^\delta }{J_p}\left( {\frac{1}{u}} \right){\rm{d}}u} = \mathit{\Gamma }\left( {\delta + 1} \right){J_{\delta + p + 1}}\left( {\frac{1}{t}} \right) $ (9)

    类似于定理1的证明方法,可得结论成立.

    定理3  若r(F)=0,定义

    $ {\mu _p}\left( t \right) = E\left\{ {{{\left( {\left( {\log \left| { - X} \right|} \right) + t} \right)}^p}\left| {\left( {\log \left| { - X} \right|} \right) > - t} \right.} \right\} $

    如果FDp(H3α),α>2,那么对于0≤pα-2,μp+2(t)<∞且t→∞时,

    $ \frac{{{\mu _p}\left( t \right){\mu _{p + 2}}\left( t \right)}}{{{{\left\{ {{\mu _{p + 1}}\left( t \right)} \right\}}^2}}} \to \frac{{\left( {p + 2} \right)\left( { \alpha - p - 1} \right)}}{{\left( {p + 1} \right)\left( { \alpha - p - 2} \right)}} $ (10)

    反之,当α>2时,对0≤pα-2,如果μp+2(t)<∞且式(10)成立,那么FDp(H3α).

      必要性的证明类似于定理1中(4)式的证明,利用文献[2]中定理2.3关于FDp(H3α)的充要条件为当y>0时,

    $ \mathop {\lim }\limits_{t \to \infty } \frac{{1 - F\left( { - \exp \left( { - ty} \right)} \right)}}{{1 - F\left( { - \exp \left( { - t} \right)} \right)}} = {y^{ - \alpha }} $

    证得(10)式成立.对充分性,定义

    $ {J_p}\left( t \right) = {\left\{ {\mathit{\Gamma }\left( {p + 1} \right)} \right\}^{ - 1}}\int_{ - \exp \left( { - t} \right)}^0 {{{\left( {\left( {\log \left| { - X} \right|} \right) + t} \right)}^p}{\rm{d}}F\left( x \right)} $

    $ \int_t^\infty {{{\left( {u - t} \right)}^\delta }{J_p}\left( u \right){\rm{d}}u} = {\left( { - 1} \right)^{\delta + 1}}\mathit{\Gamma }\left( {\delta + 1} \right){J_{\delta + p + 1}}\left( t \right) $ (11)

    类似于定理1的证明方法,可得结论成立.

    定理4  若r(F)<0,定义

    $ {\mu _p}\left( t \right) = E\left\{ {{{\left( {\left( { - \log \left| {\frac{X}{{r\left( F \right)}}} \right|} \right) + \frac{1}{t}} \right)}^p}\left| {\left( { - \log \left| {\frac{X}{{r\left( F \right)}}} \right|} \right) > - \frac{1}{t}} \right.} \right\} $

    如果FDp(H4α),α>2,那么对于0≤pα-2,μp+2(t)<∞且t→∞时,

    $ \frac{{{\mu _p}\left( t \right){\mu _{p + 2}}\left( t \right)}}{{{{\left\{ {{\mu _{p + 1}}\left( t \right)} \right\}}^2}}} \to \frac{{\left( {p + 2} \right)\left( { - \alpha - p - 1} \right)}}{{\left( {p + 1} \right)\left( { - \alpha - p - 2} \right)}} $ (12)

    反之,当α>2时,对0≤pα-2,如果μp+2(t)<∞且式(12)成立,那么FDp(H4α).

      必要性的证明类似于定理1中(4)式的证明,利用文献[2]中定理2.4关于FDp(H4α)的充要条件为当y>0时

    $ \mathop {\lim }\limits_{t \to \infty } \frac{{1 - F\left( {r\left( F \right)\exp \left( {\frac{y}{t}} \right)} \right)}}{{1 - F\left( {r\left( F \right)\exp \left( {\frac{1}{t}} \right)} \right)}} = {y^\alpha } $

    可证得(12)式成立.对充分性,定义

    $ {J_p}\left( t \right) = {\left\{ {\mathit{\Gamma }\left( {p + 1} \right)} \right\}^{ - 1}}\int_{r\left( F \right)\exp \left( {\frac{1}{t}} \right)}^{r\left( F \right)} {{{\left( {\left( { - \log \left| {\frac{X}{{r\left( F \right)}}} \right|} \right) + \frac{1}{t}} \right)}^p}{\rm{d}}F\left( x \right)} $

    $ \int_t^\infty {{{\left( {u - t} \right)}^\delta }{J_p}\left( {\frac{1}{u}} \right){\rm{d}}u} = {\left( { - 1} \right)^{\delta + 1}}\mathit{\Gamma }\left( {\delta + 1} \right){J_{\delta + p + 1}}\left( {\frac{1}{t}} \right) $ (13)

    类似于定理1的证明方法,可得结论成立.

    定理5  若r(F)>0,定义

    $ {\mu _p}\left( t \right) = E\left\{ {{{\left( {\left( {\log \left| {\frac{X}{t}} \right|} \right) - f\left( t \right)} \right)}^p}\left| {\left( {\log \left| {\frac{X}{t}} \right|} \right) > f\left( t \right)} \right.} \right\} $

    如果FDp(Φ),那么对于p≥0,μp+2(t)<∞且当tr(F)时,

    $ \frac{{{\mu _p}\left( t \right){\mu _{p + 2}}\left( t \right)}}{{{{\left\{ {{\mu _{p + 1}}\left( t \right)} \right\}}^2}}} \to \frac{{p + 2}}{{p + 1}} $ (14)

    反之,对p≥0,如果μp+2(t)<∞且式(14)成立,那么FDp(Φ).

      必要性的证明:由文献[2]定理2.5可知,FDp(Φ)的等价条件为当r(F)>0时,存在一个函数f>0,使得对于x$\mathbb{R} $

    $ \mathop {\lim }\limits_{t \to r\left( F \right)} \frac{{1 - F\left( {t\exp \left( {xf\left( t \right)} \right)} \right)}}{{1 - F\left( t \right)}}\exp \left( { - x} \right) $

    成立.在此条件下,对某些连续函数f,定义

    $ f\left( t \right) = \frac{1}{{\bar F\left( t \right)}}\int_t^{r\left( F \right)} {\frac{{\bar F\left( s \right)}}{s}{\rm{d}}s} $

    $ \int_t^{r\left( F \right)} {\frac{{\bar F\left( x \right)}}{x}{\rm{d}}x} < \infty $

    因此有

    $ \begin{array}{l} {\mu _p}\left( t \right) = E\left\{ {{{\left( {\left( {\log \left| {\frac{X}{t}} \right|} \right) - f\left( t \right)} \right)}^p}\left| {\left( {\log \left| {\frac{X}{t}} \right|} \right) > f\left( t \right)} \right.} \right\} = \\ \;\;\;\;\;\;\;\;\;\;\;p{\left( {f\left( t \right)} \right)^p}\int_1^{\frac{{\log \left| {\frac{{r\left( F \right)}}{t}} \right|}}{{f\left( t \right)}}} {\frac{{1 - F\left( {t\exp \left( {yf\left( t \right)} \right)} \right)}}{{1 - F\left( {t\exp \left( {f\left( t \right)} \right)} \right)}}{{\left( {y - 1} \right)}^{p - 1}}{\rm{d}}y} \sim \\ \;\;\;\;\;\;\;\;\;\;\;\mathit{\Gamma }\left( {p + 1} \right){\left( {f\left( t \right)} \right)^p},t \uparrow r\left( F \right) \end{array} $ (15)

    f(t)的表达式可知

    $ \frac{{\log \left| {\frac{{r\left( F \right)}}{t}} \right|}}{{f\left( t \right)}} - 1 \to \infty \left( {t \uparrow r\left( F \right)} \right) $

    故(15)式中的渐近式成立.显然,式(15)对p=0也成立.由式(15),可证式(14)成立.

    对于充分性的证明,首先定义

    $ {J_p}\left( t \right) = {\left\{ {\mathit{\Gamma }\left( {p + 1} \right)} \right\}^{ - 1}}\int_{t\exp \left( {f\left( t \right)} \right)}^{r\left( F \right)} {{{\left( {\left( {\log \left| {\frac{X}{t}} \right|} \right) - f\left( t \right)} \right)}^p}{\rm{d}}F\left( x \right)} \;\;\;\;\;\;\;\;\;\;t < r\left( F \right) $

    另外记

    $ {g_1}\left( t \right) = \log \left( t \right) + f\left( t \right) $

    类似于式(5)的方法,可得

    $ \int_t^\infty {{{\left( {u - {g_1}\left( t \right)} \right)}^\delta }{J_p}\left( {g_1^ \leftarrow \left( u \right)} \right){\rm{d}}u} = \mathit{\Gamma }\left( {\delta + 1} \right){J_{\delta + p + 1}}\left( {{g_1}\left( t \right)} \right) $ (16)

    类似于定理1的证明过程,可得结论成立.

    定理6  若r(F)≤0,定义

    $ {\mu _p}\left( t \right) = E\left\{ {{{\left( {\left( {\log \left| {\frac{X}{t}} \right|} \right) + f\left( t \right)} \right)}^p}\left| {\left( {\log \left| {\frac{X}{t}} \right|} \right) > - f\left( t \right)} \right.} \right\} $

    如果FDp(Ψ),那么对于p≥0,μp+2(t)<∞且当tr(F)时,

    $ \frac{{{\mu _p}\left( t \right){\mu _{p + 2}}\left( t \right)}}{{{{\left\{ {{\mu _{p + 1}}\left( t \right)} \right\}}^2}}} \to \frac{{p + 2}}{{p + 1}} $ (17)

    反之,对p≥0,如果μp+2(t)<∞且(17)式成立,那么FDp(Ψ).

      必要性的证明:由文献[2]定理2.6可知,FDp(Ψ)的等价条件为当r(F)≤0时,存在一个函数f>0使得对于x$\mathbb{R} $,有

    $ \mathop {\lim }\limits_{t \to r\left( F \right)} \frac{{1 - F\left( {t\exp \left( {xf\left( t \right)} \right)} \right)}}{{1 - F\left( t \right)}} = \exp \left( x \right) $

    成立.在此条件下,对某些连续函数f,定义

    $ f\left( t \right) = - \frac{1}{{\bar F\left( t \right)}}\int_t^{r\left( F \right)} {\frac{{\bar F\left( s \right)}}{s}{\rm{d}}s} $

    $ - \int_t^{r\left( F \right)} {\frac{{\bar F\left( x \right)}}{x}{\rm{d}}x} < \infty $

    成立.因此有,

    $ \begin{array}{*{20}{c}} {{\mu _p}\left( t \right) \sim \mathit{\Gamma }\left( {p + 1} \right){{\left( { - f\left( t \right)} \right)}^p}}&{r \uparrow r\left( F \right)} \end{array} $ (18)

    由(18)式,可证(17)式成立.

    对于充分性的证明,首先定义

    $ \begin{array}{*{20}{c}} {{J_p}\left( t \right) = {{\left\{ {\mathit{\Gamma }\left( {p + 1} \right)} \right\}}^{ - 1}}\int_{t\exp \left( { - f\left( t \right)} \right)}^{r\left( F \right)} {{{\left( {\left( {\log \left| {\frac{x}{t}} \right|} \right) + f\left( t \right)} \right)}^p}{\rm{d}}F\left( x \right)} }&{t < r\left( F \right)} \end{array} $

    类似于定理5,记g2(t)=log (t)-f(t),则

    $ \int_t^\infty {{{\left( {u - {g_2}\left( t \right)} \right)}^\delta }{J_p}\left( {g_2^ \leftarrow \left( u \right)} \right){\rm{d}}u} = \mathit{\Gamma }\left( {\delta + 1} \right){J_{\delta + p + 1}}\left( {{g_2}\left( t \right)} \right) $ (19)

    同理可证得结论成立.

    参考文献
    [1] PANTCHEVA E. Limit Theorems for Extreme Order Statistics Under Nonlinear Normalization[M]//KALASHNIKOV V V, ZOLOTATEV V M. Stability Problems for Stochastic Models. Berlin: Springer, 1985: 284-309.
    [2] MOHAN N R, RAVI S. Max Domains of Attraction of Univariate and Multivariate P-Max Stable Laws[J]. Theory of Probability and Its Applications, 1993, 37(4): 632-643. DOI:10.1137/1137119
    [3] PENG Z X, SHUAI Y L, NADARAJAH S. On Convergence of Extremes Under Power Normalization[J]. Extremes, 2013, 16(3): 285-301. DOI:10.1007/s10687-012-0161-2
    [4] RESNICK S I. Extreme Values, Regular Variation, and Point Processes[M]. New York: Springer-Verlag, 1987.
    [5] GELUK J L. On The Domain of Attraction of exp (-exp (-x))[J]. Statistics and Probability Letters, 1996, 31(2): 91-95.
    Conditions Based on Conditional Moments for p-Max Stable Laws Conditional Moment Characterization of Limit Distribution Under Power Normalization
    PENG Xi, ZHOU Wei, PENG Zuo-xiang     
    School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
    Abstract: Let {Xn} be independent and identically distributed random variables with the common distribution function F(x). Necessary and sufficient conditions for F belonging to the domains of attraction of H1, α, H2, α, H3, α, H4, α, ψ(x) and Ф(x) with nondegenerate univariate marginals under power normalization are derived in terms of conditional moments.
    Key words: limit distribution under power normalization    conditional moment    domain of attraction    
    X