西南大学学报 (自然科学版)  2018, Vol. 40 Issue (11): 61-66.  DOI: 10.13718/j.cnki.xdzk.2018.11.010
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  • 平稳序列的次最大值和次最小值与其位置的渐近性质    [PDF全文]
    卜雪妍, 陈守全     
    西南大学 数学与统计学院, 重庆 400715
    摘要:讨论了含有极值指标的平稳序列的次最大值与其位置在长相依条件下的渐近性质,并给出了次最大值和次最小值与它们的位置的渐近性质.
    关键词点过程    相依条件    最大值的位置    渐近独立    

    极值理论的研究在人类生活中具有非常重要的地位和作用,常用于预测一些极端随机现象和小概率事件.如今,由于极值理论的应用日益成熟、理论本身的发展以及经济金融领域频繁发生危机,对人类的生活和社会造成了重大影响,使得极值理论的价值和优越性进一步得以体现,逐步走向了金融风险管理领域.而且,极值统计量和它们的位置关系的研究作为极值理论研究的一部分,在极值分析中有着非常重要的作用,主要被应用在对环境和财经的数据处理分析中,当我们所研究的数据部分丢失时,应怎样选取数据确保这组数据达到我们预期的概率.类似的问题促使研究者们对极值顺序统计量位置的渐进分布进行深入研究.

    对于平稳序列,文献[1]已经证明了最大值和其首次出现的位置在独立同分布的情况下渐近独立以及最大值的位置渐近服从均匀分布;文献[2]得到了在弱混合条件下,平稳序列的最大值与最小值位置的联合渐近分布;文献[3]讨论了平稳高斯向量序列最大值与最小值联合的几乎处处收敛;文献[4]讨论了高斯三角阵的最大值与最小值的密度函数的渐近性质.到目前为止,对于次最大值与次最小值的渐近性质尚未研究,因此,本文在已有研究的基础上对平稳序列的次最大值和它的位置的渐近性质以及次最大值和次最小值位置的渐近性质作进一步研究.

    设{Xn}n≥1是严格的平稳序列,且存在实数列{an>0}n≥1和{bn},使得对于任意的x$\mathbb{R}$

    $ P\left\{ {a_n^{ - 1}\left( {{M_n} - {b_n}} \right) \le x} \right\} \to G\left( x \right) $ (1)

    其中G表示非退化的分布函数.如果对每个τ>0,存在{un(τ)}n≥1使得

    $ \mathop {\lim }\limits_{n \to \infty } n\left( {1 - F\left( {u_n^{\left( \tau \right)}} \right)} \right) \to \tau $

    $ \mathop {\lim }\limits_{n \to \infty } P\left( {{M_n} \le u_n^{\left( \tau \right)}} \right) \to \exp \left\{ { - \theta \tau } \right\} $

    成立,那么就说{Xn}有极值指标θ,0<θ≤1,因此(1)式成立当且仅当

    $ \mathop {\lim }\limits_{n \to \infty } n\left( {1 - F\left( {{u_n}\left( x \right)} \right)} \right) \to \tau \left( x \right) = - \frac{1}{\theta }\log G\left( x \right),x \in \mathbb{R} $

    Mn(k)(I)和mn(r)(I)分别表示该序列的第k个最大值和第r个最小值,$\bar L_n^{\left( k \right)}$$\underline L _n^{\left( r \right)}$分别表示第k个最大值的位置和第r个最小值的位置,即:

    $ \bar L_n^{\left( k \right)} = \min \left\{ {1 \leqslant j \leqslant n:M_n^{\left( k \right)} = {X_j}} \right\} $
    $ \underline L _n^{\left( r \right)} = \min \left\{ {1 \leqslant j \leqslant n:m_n^{\left( r \right)} = {X_j}} \right\} $
    1 预备知识

    下面先定义平稳序列超过水平un(x)=anx+bn所构成的点过程Nn

    $ {N_n}\left( \mathbb{B} \right) = \sum\limits_{i = 1}^n {{I_{\left\{ {i/n \in \mathbb{B},{x_i} > {u_n}\left( x \right)} \right\}}}} $

    其中$\mathbb{B}$是(0,1]上的Borel集.根据上述定义显然有

    $ P\left\{ {N\left( \mathbb{B} \right) = 0} \right\} = P\left\{ {{M_n}\left( \mathbb{B} \right) \leqslant {u_n}} \right\} $

    如果对于任意的τn(1-F(un))=np{xiun}→τ,文献[1]中的条件D(un(τ))和D(un(τ))成立,那么点过程Nn收敛到参数为τ的泊松过程N.

    文献[1]中对平稳序列在相依条件下的最大值分布做了研究,文献[5-6]证明了平稳序列在相依条件下最大值和最小值是渐近独立的.下面将对文献[1, 5-6]的条件进行推广,使得{Mn(1)(I)≤un(1)Mn(2)(I)≤un(2)}和{Mn(1)(J)≤un(3)Mn(2)(J)≤un(4)}渐近独立.对于{1,2,…,n}中的子集IJEF且有IJ={1,2,…,n},EF={1,2,…,n}和IJ=EF=,使得{{Mn(1)(I)≤un(1)Mn(2)(I)≤un(2)Mn(1)(J)≤un(3)Mn(2)(J)}≤un(4)}和{mn(1)(E)>vn(1)mn(2)(E)>vn(2)mn(1)(F)>vn(3)mn(2)(F)>vn(4)}渐近独立,其中un(k)vn(k)k=1,2,3,4满足

    $ n\left( {1 - F\left( {u_n^{\left( k \right)}} \right)} \right) \to {\tau _k},n\left( {F\left( {v_n^{\left( k \right)}} \right)} \right) \to {\eta _k} $

    定义1  如果对于给定的实数序列{(un(1)un(2)un(3)un(4))},且un(i)*∈{(un(1)un(2)un(3)un(4))}使得

    $ \begin{array}{l} {\alpha _{n,l}} = \mathop {\sup }\limits_{\begin{array}{*{20}{c}} {1 \le {i_1} < \cdots < {i_p} < {j_1} < \cdots < {j_q} \le n}\\ {{j_1} - {i_p} > l} \end{array}} \left| {P\left( {\bigcap\limits_{s = 1}^p {\left\{ {{X_{{i_s}}} \le u_{n{i_s}}^ * } \right\}} ,\bigcap\limits_{s = 1}^q {\left\{ {{X_{{j_s}}} \le u_{n{j_s}}^ * } \right\}} } \right)} \right. - \\ \;\;\;\;\;\;\;\;\;\left. {P\left( {\bigcap\limits_{s = 1}^p {\left\{ {{X_{{i_s}}} \le u_{n{i_s}}^ * } \right\}} } \right)P\left( {\bigcap\limits_{s = 1}^q {\left\{ {{X_{{j_s}}} \le u_{n{j_s}}^ * } \right\}} } \right)} \right| \to 0 \end{array} $

    其中ln=o(n).那么就说序列{Xn}n≥1满足条件D4(un(1)un(2)un(3)un(4)).

    定义2  假设{un(k)}n≥1,{vn(k)}n≥1k=1,2,3,4是两个实数序列,如果

    $ \begin{array}{l} \alpha _{n,l}^{\left( {{1^ * }} \right)} = \mathop {\sup }\limits_{\begin{array}{*{20}{c}} {1 \le {i_1} < \cdots < {i_p} < {j_1} < \cdots < {j_q} \le n}\\ {{j_1} - {i_p} > l} \end{array}} \left| {P\left( {\bigcap\limits_{s = 1}^p {\left\{ {{X_{{i_s}}} \le u_n^{\left( {{i_s}} \right) * }} \right\}} ,\bigcap\limits_{s = 1}^q {\left\{ {{X_{{j_s}}} \le u_n^{\left( {{i_s}} \right) * }} \right\}} } \right)} \right. - \\ \;\;\;\;\;\;\;\;\;\left. {P\left( {\bigcap\limits_{s = 1}^p {\left\{ {{X_{{i_s}}} \le u_n^{\left( {{i_s}} \right) * }} \right\}} } \right)P\left( {\bigcap\limits_{s = 1}^q {\left\{ {{X_{{j_s}}} \le u_n^{\left( {{i_s}} \right) * }} \right\}} } \right)} \right| \end{array} $
    $ \begin{array}{l} \alpha _{n,l}^{\left( {{2^ * }} \right)} = \mathop {\sup }\limits_{\begin{array}{*{20}{c}} {1 \le {i_1} < \cdots < {i_p} < {j_1} < \cdots < {j_q} \le n}\\ {{j_1} - {i_p} > l} \end{array}} \left| {P\left( {\bigcap\limits_{s = 1}^p {\left\{ {{X_{{i_s}}} > v_n^{\left( {{i_s}} \right) * }} \right\}} ,\bigcap\limits_{s = 1}^q {\left\{ {{X_{{j_s}}} > v_n^{\left( {{i_s}} \right) * }} \right\}} } \right)} \right. - \\ \;\;\;\;\;\;\;\;\;\left. {P\left( {\bigcap\limits_{s = 1}^p {\left\{ {{X_{{i_s}}} > v_n^{\left( {{i_s}} \right) * }} \right\}} } \right)P\left( {\bigcap\limits_{s = 1}^q {\left\{ {{X_{{j_s}}} > v_n^{\left( {{i_s}} \right) * }} \right\}} } \right)} \right| \end{array} $
    $ \begin{array}{l} \alpha _{n,l}^{\left( {{3^ * }} \right)} = \mathop {\sup }\limits_{\begin{array}{*{20}{c}} {1 \le {i_1} < \cdots < {i_p} < {j_1} < \cdots < {j_q} \le n}\\ {{j_1} - {i_p} > l} \end{array}} \left| {P\left( {\bigcap\limits_{s = 1}^p {\left\{ {v_n^{\left( {{i_s}} \right) * } < {X_{{i_s}}} \le u_n^{\left( {{i_s}} \right) * }} \right\}} ,\bigcap\limits_{s = 1}^q {\left\{ {v_n^{\left( {{i_s}} \right) * } < {X_{{j_s}}} \le u_n^{\left( {{i_s}} \right) * }} \right\}} } \right)} \right. - \\ \;\;\;\;\;\;\;\;\;\left. {P\left( {\bigcap\limits_{s = 1}^p {\left\{ {v_n^{\left( {{i_s}} \right) * } < {X_{{i_s}}} \le u_n^{\left( {{i_s}} \right) * }} \right\}} } \right)P\left( {\bigcap\limits_{s = 1}^q {\left\{ {v_n^{\left( {{i_s}} \right) * } < {X_{{j_s}}} \le u_n^{\left( {{i_s}} \right) * }} \right\}} } \right)} \right| \end{array} $
    $ \alpha _{n,l}^ * = \mathop {\sup }\limits_{i = 1,2,3} \alpha _{n,l}^{{{\left( i \right)}^ * }} \to 0 $

    其中ln=o(n),un(i)*∈{un(1)un(2)un(3)un(4)},vn(j)*∈{vn(1)vn(2)vn(3)vn(4)}.那么我们就说序列{Xn}n≥1满足条件D((un(1)un(2)un(3)un(4)),(vn(1)vn(2)vn(3)vn(4))).

    定义3  如果

    $ \mathop {\lim }\limits_{n \to \infty } \sup {k_n}\sum\limits_{i = 1}^{\left[ {n/{k_n}} \right]} {\sum\limits_{j = 1}^{\left[ {n/{k_n}} \right]} {P\left\{ {{X_i} > u_n^{{{\left( i \right)}^ * }},{X_j} > v_n^{{{\left( j \right)}^ * }}} \right\}} } = 0 $

    其中un(i)*∈{un(1)un(2)un(3)un(4)},vn(j)*∈{vn(1)vn(2)vn(3)vn(4)}. {kn}n≥1是一个整数序列,且当n→∞时,kn→∞,那么序列{Xn}n≥1满足条件C2((un(1)un(2)un(3)un(4)),(vn(1)vn(2)vn(3)vn(4))).

    对任意的IRn={1,…,n},JRn={1,…,n},IJ=RnIJ=,下面的引理说明了在条件D4(un(1)un(2)un(3)un(4))下{Mn(1)(I)≤un(1)Mn(2)(I)≤un(2)}和{Mn(1)(J)≤un(3)Mn(2)(J)≤un(4)}渐近独立,这是得到次最大值和它的位置的渐近性质的关键.

    引理1  假设平稳序列{Xn}n≥1满足条件D4(un(1)un(2)un(3)un(4)),且存在整数序列{kn}n≥1和{ln}n≥1使得

    $ {k_n}{l_n}/n \to 0{k_n}\alpha \left\{ {n,{l_n}} \right\} \to 0 $

    其中αnl是条件D4(un(1)un(2)un(3)un(4))的系数,则有

    $ \begin{array}{*{20}{c}} {\mathop {\lim }\limits_{n \to \infty } P\left\{ {\left\{ {M_n^{\left( 1 \right)}\left( I \right) \le u_n^{\left( 1 \right)},M_n^{\left( 2 \right)}\left( I \right) \le u_n^{\left( 2 \right)},M_n^{\left( 1 \right)}\left( J \right) \le u_n^{\left( 3 \right)},M_n^{\left( 2 \right)}\left( J \right)} \right\} \le u_n^{\left( 4 \right)}} \right\} - }\\ {P\left\{ {M_n^{\left( 1 \right)}\left( I \right) \le u_n^{\left( 1 \right)},M_n^{\left( 2 \right)}\left( I \right) \le u_n^{\left( 2 \right)}} \right\}P\left\{ {M_n^{\left( 1 \right)}\left( J \right) \le u_n^{\left( 3 \right)},M_n^{\left( 2 \right)}\left( J \right) \le u_n^{\left( 4 \right)}} \right\} = 0} \end{array} $

    引理2  假设{un(k)}n≥1,{vn(k)}n≥1k=1,2,3,4是实数序列,如果对于k∈{1,2,3,4},有

    $ n\left( {1 - F\left( {u_n^{\left( k \right)}} \right)} \right) \to {\tau _k},n\left( {F\left( {v_n^{\left( k \right)}} \right)} \right) \to {\eta _k} $ (2)

    其中τkηk<∞.且平稳序列{Xn}n≥1满足条件αnln*=o(1),整数序列{kn}n≥1满足

    $ {k_n}{l_n}/n \to 0,{k_n}\alpha _{n,{l_n}}^ * \to 0,{k_n} \to \infty $ (3)

    其中J1J2,…,J{kn}为{1,2,…,n}中不相交的子集,则有

    $ \begin{array}{*{20}{c}} {\mathop {\lim }\limits_{n \to \infty } P\left( {\bigcap\limits_{i = 1}^{{k_n}} {\left\{ {v_n^{{{\left( i \right)}^ * }} < m_n^{\left( 2 \right)}\left( {{J_i}} \right) < M_n^{\left( 2 \right)}\left( {{J_i}} \right) \le u_n^{{{\left( i \right)}^ * }}} \right\}} } \right) - }\\ {\prod\limits_{i = 1}^{{k_n}} {P\left( {v_n^{{{\left( i \right)}^ * }} < m_n^{\left( 2 \right)}\left( {{J_i}} \right) < M_n^{\left( 2 \right)}\left( {{J_i}} \right) \le u_n^{{{\left( i \right)}^ * }}} \right)} = 0} \end{array} $

    其中un(i)*∈{un(1)un(2)un(3)un(4)},vn(j)*∈{vn(1)vn(2)vn(3)vn(4)}.

    引理3  假设序列{Xn}n≥1和{-Xn}n≥1分别有极值指标θ1θ2,且平稳序列{Xn}n≥1满足条件D((un(1)un(2)un(3)un(4)),(vn(1)vn(2)vn(3)vn(4)))和C2((un(1)un(2)un(3)un(4)),(vn(1)vn(2)vn(3)vn(4)))其中un(k)vn(k)满足(2)式,{kn}n≥1满足(3)式,则

    $ \begin{array}{*{20}{c}} {\mathop {\lim }\limits_{n \to \infty } P\left( {M_n^{\left( 1 \right)}\left( I \right) \le u_n^{\left( 1 \right)},M_n^{\left( 2 \right)}\left( I \right) \le u_n^{\left( 2 \right)},M_n^{\left( 1 \right)}\left( J \right) \le u_n^{\left( 3 \right)},M_n^{\left( 2 \right)}\left( J \right) \le u_n^{\left( 4 \right)},} \right.}\\ {\left. {m_n^{\left( 1 \right)}\left( E \right) > v_n^{\left( 1 \right)},m_n^{\left( 2 \right)}\left( E \right) > v_n^{\left( 2 \right)},m_n^{\left( 1 \right)}\left( F \right) > v_n^{\left( 3 \right)},m_n^{\left( 2 \right)}\left( F \right) > v_n^{\left( 4 \right)}} \right) - }\\ {P\left( {M_n^{\left( 1 \right)}\left( I \right) \le u_n^{\left( 1 \right)},M_n^{\left( 2 \right)}\left( I \right) \le u_n^{\left( 2 \right)},M_n^{\left( 1 \right)}\left( J \right) \le u_n^{\left( 3 \right)},M_n^{\left( 2 \right)}\left( J \right) \le u_n^{\left( 4 \right)}} \right) \times }\\ {P\left( {m_n^{\left( 1 \right)}\left( E \right) > v_n^{\left( 1 \right)},m_n^{\left( 2 \right)}\left( E \right) > v_n^{\left( 2 \right)},m_n^{\left( 1 \right)}\left( F \right) > v_n^{\left( 3 \right)},m_n^{\left( 2 \right)}\left( F \right) > v_n^{\left( 4 \right)}} \right) = 0} \end{array} $

      不妨设t1t2,取I1={1,…,[nt1]},I2={[nt1]+1,…,[nt2]}和I3={[nt2]+1,…,n},则利用引理2,

    $ \begin{array}{*{20}{c}} {\mathop {\lim }\limits_{n \to \infty } P\left( {M_n^{\left( 1 \right)}\left( I \right) \le u_n^{\left( 1 \right)},M_n^{\left( 2 \right)}\left( I \right) \le u_n^{\left( 2 \right)},M_n^{\left( 1 \right)}\left( J \right) \le u_n^{\left( 3 \right)},M_n^{\left( 2 \right)}\left( J \right) \le u_n^{\left( 4 \right)},} \right.}\\ {\left. {m_n^{\left( 1 \right)}\left( E \right) > v_n^{\left( 1 \right)},m_n^{\left( 2 \right)}\left( E \right) > v_n^{\left( 2 \right)},m_n^{\left( 1 \right)}\left( F \right) > v_n^{\left( 3 \right)},m_n^{\left( 2 \right)}\left( F \right) > v_n^{\left( 4 \right)}} \right)}\\ { = \mathop {\lim }\limits_{n \to \infty } P\left( {v_n^{\left( 1 \right)} < m_n^{\left( 1 \right)}\left( {{I_1}} \right) < m_n^{\left( 2 \right)}\left( {{I_1}} \right) < M_n^{\left( 2 \right)}\left( {{I_1}} \right) < M_n^{\left( 1 \right)}\left( {{I_1}} \right) < u_n^{\left( 1 \right)}} \right) \times }\\ {P\left( {v_n^{\left( 1 \right)} < m_n^{\left( 1 \right)}\left( {{I_2}} \right) < m_n^{\left( 2 \right)}\left( {{I_2}} \right) < M_n^{\left( 2 \right)}\left( {{I_2}} \right) < M_n^{\left( 1 \right)}\left( {{I_2}} \right) < u_n^{\left( 2 \right)}} \right) \times }\\ {P\left( {v_n^{\left( 2 \right)} < m_n^{\left( 1 \right)}\left( {{I_3}} \right) < m_n^{\left( 2 \right)}\left( {{I_3}} \right) < M_n^{\left( 2 \right)}\left( {{I_3}} \right) < M_n^{\left( 1 \right)}\left( {{I_3}} \right) < u_n^{\left( 2 \right)}} \right)} \end{array} $ (4)

    由题设条件易知,对于任意的k∈{1,2,3,4},C2(un(k)vn(k))和D(un(k)vn(k))成立,则(4)式等于

    $ \begin{array}{*{20}{c}} {\mathop {\lim }\limits_{n \to \infty } P\left( {v_n^{\left( 1 \right)} < m_n^{\left( 1 \right)}\left( {{I_1}} \right) < m_n^{\left( 2 \right)}\left( {{I_1}} \right)} \right) \times P\left( {M_n^{\left( 2 \right)}\left( {{I_1}} \right) < M_n^{\left( 1 \right)}\left( {{I_1}} \right) < u_n^{\left( 1 \right)}} \right) \times }\\ {P\left( {v_n^{\left( 1 \right)} < m_n^{\left( 1 \right)}\left( {{I_2}} \right) < m_n^{\left( 2 \right)}\left( {{I_2}} \right)} \right) \times P\left( {M_n^{\left( 2 \right)}\left( {{I_2}} \right) < M_n^{\left( 1 \right)}\left( {{I_2}} \right) < u_n^{\left( 2 \right)}} \right) \times }\\ {P\left( {v_n^{\left( 2 \right)} < m_n^{\left( 1 \right)}\left( {{I_3}} \right) < m_n^{\left( 2 \right)}\left( {{I_3}} \right)} \right) \times P\left( {M_n^{\left( 2 \right)}\left( {{I_3}} \right) < M_n^{\left( 1 \right)}\left( {{I_3}} \right) < u_n^{\left( 2 \right)}} \right)} \end{array} $ (5)

    因此,利用平稳性以及序列{Xn}n≥1和{-Xn}n≥1分别有极值指标θ1θ2,得到(5)式等于

    $ \begin{array}{*{20}{c}} {\left( {{t_1}\left( {{\tau _2} - {\tau _1} + 1} \right)} \right)\left( {\left( {1 - {t_1}} \right)\left( {{\tau _4} - {\tau _3}} \right) + 1} \right)\left( {{t_2}\left( {{\eta _2} - {\eta _1} + 1} \right)} \right)\left( {\left( {1 - {t_2}} \right)\left( {{\eta _4} - {\eta _3}} \right) + 1} \right) \times }\\ {\exp \left( { - {\theta _1}{t_1}{\tau _2} - {\theta _1}\left( {1 - {t_1}} \right){\tau _4} - {\theta _2}{t_2}{\eta _2} - {\theta _2}\left( {1 - {t_2}} \right){\eta _4}} \right)} \end{array} $ (6)

    另一方面,由于{Xn}n≥1和{-Xn}n≥1分别满足条件D4(un(1)un(2)un(3)un(4))和D4(-vn(1),-vn(2),-vn(3),-vn(4)),利用引理1,

    $ \begin{array}{*{20}{c}} {P\left( {M_n^{\left( 1 \right)}\left( I \right) \le u_n^{\left( 1 \right)},M_n^{\left( 2 \right)}\left( I \right) \le u_n^{\left( 2 \right)},M_n^{\left( 1 \right)}\left( J \right) \le u_n^{\left( 3 \right)},M_n^{\left( 2 \right)}\left( J \right) \le u_n^{\left( 4 \right)}} \right) \times }\\ {P\left( {m_n^{\left( 1 \right)}\left( E \right) > v_n^{\left( 1 \right)},m_n^{\left( 2 \right)}\left( E \right) > v_n^{\left( 2 \right)},m_n^{\left( 1 \right)}\left( F \right) > v_n^{\left( 3 \right)},m_n^{\left( 2 \right)}\left( F \right) > v_n^{\left( 4 \right)}} \right) = }\\ {P\left( {M_n^{\left( 1 \right)}\left( I \right) \le u_n^{\left( 1 \right)},M_n^{\left( 2 \right)}\left( I \right) \le u_n^{\left( 2 \right)}} \right) \times P\left( {M_n^{\left( 1 \right)}\left( J \right) \le u_n^{\left( 3 \right)},M_n^{\left( 2 \right)}\left( J \right) \le u_n^{\left( 4 \right)}} \right) \times }\\ {P\left( {m_n^{\left( 1 \right)}\left( E \right) > v_n^{\left( 1 \right)},m_n^{\left( 2 \right)}\left( E \right) > v_n^{\left( 2 \right)}} \right)P\left( {m_n^{\left( 1 \right)}\left( F \right) > v_n^{\left( 3 \right)},m_n^{\left( 2 \right)}\left( F \right) > v_n^{\left( 4 \right)}} \right) + o\left( 1 \right)} \end{array} $ (7)

    因此(7)式也收敛到(6)式,从而结论成立.

    2 主要结果及其证明

    下面将给出${\overline {{L_n}} ^{\left( 2 \right)}}/n$an-1(Mn(2)-bn)的渐近性质以及次最大值和次最小值与其位置的渐近性质.

    定理1  假设{Xn}n≥1是一平稳序列,有极值指标θ,且存在常数序列{an}n≥1和{bn}n≥1使得

    $ \mathop {\lim }\limits_{n \to \infty } P\left\{ {M_n^{\left( 1 \right)} \le {a_n}x + {b_n}} \right\} \to {G^\theta }\left( x \right) $

    其中G是非退化的分布函数.如果对于每个xk$\mathbb{R}$un(k)=an-1xk+bnk=1,2,3,4,且{Xn}n≥1满足条件D4(un(1)un(2)un(3)un(4))和D(un(k)),则

    $ \mathop {\lim }\limits_{n \to \infty } P\left\{ {{{\overline {{L_n}} }^{\left( 2 \right)}}/n \le t,a_n^{\left( { - 1} \right)}\left( {M_n^{\left( 2 \right)} - {b_n}} \right) \le x} \right\} \to t{G^\theta }\left( x \right)\left( {1 - \log G\left( x \right)} \right) $ (8)

    其中x是实数,0<t≤1.

    也就是说,次最大值和其位置是渐近独立的,且位置是渐近地服从均匀分布.

      由于

    $ n\left( {1 - F\left( {u_n^{\left( k \right)}} \right)} \right) \to {\tau _k} $

    所以

    $ {\tau _k} = - \log G\left( {{x_k}} \right) $

    I={1,2,…,[nt]},J={[nt]+1,…,n},0<t≤1;令M(1)(I),M(2)(I)分别表示区间I上的最大值和第二最大值;令M(1)(J),M(2)(J)分别表示区间J上的最大值和第二最大值.设

    $ \begin{array}{*{20}{c}} {X_n^{\left( 1 \right)} = {a_n}\left( {M_n^{\left( 1 \right)}\left( I \right) - {b_n}} \right)}&{X_n^{\left( 2 \right)} = {a_n}\left( {M_n^{\left( 2 \right)}\left( I \right) - {b_n}} \right)} \end{array} $
    $ \begin{array}{*{20}{c}} {Y_n^{\left( 1 \right)} = {a_n}\left( {M_n^{\left( 1 \right)}\left( J \right) - {b_n}} \right)}&{Y_n^{\left( 2 \right)} = {a_n}\left( {M_n^{\left( 2 \right)}\left( J \right) - {b_n}} \right)} \end{array} $

    的联合分布函数为Hn(x1x2x3x4),则对于任意的x1x2x3x4

    $ {H_n}\left( {{x_1},{x_2},{x_3},{x_4}} \right) = P\left\{ {M_n^{\left( 1 \right)}\left( I \right) \le u_n^{\left( 1 \right)},M_n^{\left( 2 \right)}\left( I \right) \le u_n^{\left( 2 \right)},M_n^{\left( 1 \right)}\left( J \right) \le u_n^{\left( 3 \right)},M_n^{\left( 2 \right)}\left( J \right) \le u_n^{\left( 4 \right)}} \right\} $

    其中un(k)=an-1xk+bn.易知

    $ {H_n}\left( {{x_1},{x_2},{x_3},{x_4}} \right) = P\left\{ {N_n^{\left( 1 \right)}\left( {I'} \right) = 0,N_n^{\left( 2 \right)}\left( {I'} \right) \le 1,N_n^{\left( 3 \right)}\left( {J'} \right) = 0,N_n^{\left( 4 \right)}\left( {J'} \right) \le 1} \right\} $

    其中I=(0,t],J=(t,1].利用引理1和文献[1]中的推论5.5.2,取B1=IB2=J,则

    $ \begin{array}{*{20}{c}} {\mathop {\lim }\limits_{n \to \infty } {H_n}\left( {{x_1},{x_2},{x_3},{x_4}} \right) = P\left\{ {N_n^{\left( 1 \right)}\left( {I'} \right) = 0,N_n^{\left( 2 \right)}\left( {I'} \right) \le 1} \right\} \\ \times P\left\{ {N_n^{\left( 3 \right)}\left( {J'} \right) = 0,N_n^{\left( 4 \right)}\left( {J'} \right) \le 1} \right\} = }\\ {\exp \left\{ {\theta t{\tau _2}} \right\}\left( {t\left( {{\tau _2} - {\tau _1}} \right) + 1} \right)\exp \left\{ {\theta \left( {1 - t} \right){\tau _4}} \right\}\left( {\left( {1 - t} \right)\left( {{\tau _4} - {\tau _3}} \right) + 1} \right) = }\\ {{H_t}\left( {{x_1},{x_2}} \right)H\left\{ {1 - t} \right\}\left( {{x_3},{x_4}} \right) = H\left( {{x_1},{x_2},{x_3},{x_4}} \right)} \end{array} $ (9)

    其中

    $ {H_t}\left( {{x_1},{x_2}} \right) = {G^{\theta t}}\left( {{x_2}} \right)\left( {\log {G^t}\left( {{x_1}} \right) - \log {G^t}\left( {{x_2}} \right) + 1} \right) $

    由(9)式知(Xn(1)Xn(2)Yn(1)Yn(2))依分布收敛到(X1X2Y1Y2),其联合分布函数H是绝对连续的.因此

    $ \begin{array}{*{20}{c}} {\mathop {\lim }\limits_{n \to \infty } P\left\{ {{{\overline {{L_n}} }^{\left( 2 \right)}}/n \le t,{a_n}\left( {M_n^{\left( 2 \right)} - {b_n}} \right) \le {x_2}} \right\} = }\\ {\mathop {\lim }\limits_{n \to \infty } P\left\{ {M_n^{\left( 2 \right)}\left( I \right) \le u_n^{\left( 2 \right)},M_n^{\left( 2 \right)}\left( I \right) \ge M_n^{\left( 1 \right)}\left( J \right)} \right\} \\ + P\left\{ {M_n^{\left( 1 \right)}\left( I \right) \le u_n^{\left( 2 \right)},M_n^{\left( 1 \right)}\left( J \right) > M_n^{\left( 1 \right)}\left( I \right) \ge M_n^{\left( 2 \right)}\left( J \right)} \right\} = }\\ {\mathop {\lim }\limits_{n \to \infty } P\left\{ {X_n^{\left( 2 \right)} \le {x_2},X_n^{\left( 2 \right)} \le Y_n^{\left( 1 \right)}} \right\} + P\left\{ {X_n^{\left( 1 \right)} \le {x_2},X_n^{\left( 1 \right)} \le {x_2},Y_n^{\left( 1 \right)} > X_n^{\left( 1 \right)} \ge Y_n^{\left( 2 \right)}} \right\} = }\\ {P\left\{ {{X_2} \le {x_2},{X_2} \ge {Y_1}} \right\} + P\left\{ {{X_1} \le {x_2},{X_1} \le {x_2},{Y_1} > {X_1} \ge {Y_2}} \right\} = }\\ {t{G^\theta }\left( {1 - \log G\left( x \right)} \right)} \end{array} $

    定理2  假设平稳序列{Xn}n≥1和{-Xn}n≥1的极值指标分别为θ1θ2.且存在常数序列{an>0}n≥1,{bn}n≥1,{cn>0}n≥1和{dn}n≥1使得

    $ \mathop {\lim }\limits_{n \to \infty } P\left\{ {M_n^{\left( 1 \right)} \le {a_n}{x_1} + {b_n}} \right\} \to {G^{{\theta _1}}}\left( {{x_1}} \right) $

    $ \mathop {\lim }\limits_{n \to \infty } P\left\{ {m_n^{\left( 1 \right)} \le {c_n}{y_1} + {d_n}} \right\} \to 1 - {\left( {1 - H\left( {{y_1}} \right)} \right)^{{\theta _2}}} $

    成立,其中GH为非退化的分布函数.如果对于每个x1x2y1y2$\mathbb{R}$u{(k)}n=anxk+bnv{(k)}n=cnxk+dnk∈{1,2,3,4},{Xn}n≥1满足条件D((un(1)un(2)un(3)un(4)),(vn(1)vn(2)vn(3)vn(4)))和C2((un(1)un(2)un(3)un(4)),(vn(1)vn(2)vn(3)vn(4))),且整数序列{kn}n≥1满足(3)式,那么

    $ \begin{array}{*{20}{c}} {\mathop {\lim }\limits_{n \to \infty } P\left\{ {{{\overline {{L_n}} }^{\left( 2 \right)}}/n \le {t_1},a_n^{ - 1}\left( {M_n^{\left( 2 \right)} - {b_n}} \right) \le {x_2},{{\underline {{L_n}} }^{\left( 2 \right)}}/n > {t_2},a_n^{ - 1}\left( {m_n^{\left( 2 \right)} - {b_n}} \right) > {y_2}} \right\} \to }\\ {{t_1} - \left( {1 - {t_2}} \right){G^{{\theta _1}}}\left( {{x_2}} \right)\left( {1 - \log G\left( {{x_2}} \right)} \right){{\left( {1 - H\left( {{y_2}} \right)} \right)}^{{\theta _2}}}\left( {1 - \log \left( {1 - H\left( {{y_2}} \right)} \right)} \right.} \end{array} $ (10)

      由于n(1-F(un(k)))→τkn(F(vn(k)))→ηk,所以

    $ \begin{array}{*{20}{c}} {{\tau _k} = - \log G\left( {{x_k}} \right)}&{{\eta _k} = - \log \left( {1 - H\left( {{y_1}} \right)} \right)} \end{array} $

    IJEF分别表示区间{1,2,…,[nt1]},{[nt1]+1,…,n},{1,2,…,[nt2]}和{[nt2]+1,…,n};M(1)(I),M(2)(I),M(1)(J),M(2)(J)分别表示Xi中在区间IJ上的最大值和第二最大值;m(1)(E),m(2)(E),m(1)(F),m(2)(F)表示Xi中在区间EF上的最小值和第二最小值.设随机变量

    $ \begin{array}{*{20}{c}} {X_n^{\left( 1 \right)} = {a_n}\left( {M_n^{\left( 1 \right)}\left( I \right) - {b_n}} \right)}&{X_n^{\left( 2 \right)} = {a_n}\left( {M_n^{\left( 2 \right)}\left( I \right) - {b_n}} \right)} \end{array} $
    $ \begin{array}{*{20}{c}} {Y_n^{\left( 1 \right)} = {a_n}\left( {M_n^{\left( 1 \right)}\left( J \right) - {b_n}} \right)}&{Y_n^{\left( 2 \right)} = {a_n}\left( {M_n^{\left( 2 \right)}\left( J \right) - {b_n}} \right)} \end{array} $
    $ \begin{array}{*{20}{c}} {X{'}_n^{\left( 1 \right)} = {c_n}\left( {M_n^{\left( 1 \right)}\left( E \right) - {d_n}} \right)}&{X{'}_n^{\left( 2 \right)} = {c_n}\left( {M_n^{\left( 2 \right)}\left( E \right) - {d_n}} \right)} \end{array} $
    $ \begin{array}{*{20}{c}} {Y{'}_n^{\left( 1 \right)} = {c_n}\left( {M_n^{\left( 1 \right)}\left( F \right) - {d_n}} \right)}&{Y{'}_n^{\left( 2 \right)} = {c_n}\left( {M_n^{\left( 2 \right)}\left( F \right) - {d_n}} \right)} \end{array} $

    的分布函数为Hn(x1x2x3x4y1y2y3y4),则对任意的x1x2x3x4y1y2y3y4

    $ \begin{array}{*{20}{c}} {{{H'}_n}\left( {{x_1},{x_2},{x_3},{x_4},{y_1},{y_2},{y_3},{y_4}} \right) = }\\ {P\left( {M_n^{\left( 1 \right)}\left( I \right) \le u_n^{\left( 1 \right)},M_n^{\left( 2 \right)}\left( I \right) \le u_n^{\left( 2 \right)},M_n^{\left( 1 \right)}\left( J \right) \le u_n^{\left( 3 \right)},M_n^{\left( 2 \right)}\left( J \right) \le u_n^{\left( 4 \right)},} \right.}\\ {\left. {m_n^{\left( 1 \right)}\left( E \right) > v_n^{\left( 1 \right)},m_n^{\left( 2 \right)}\left( E \right) > v_n^{\left( 2 \right)},m_n^{\left( 1 \right)}\left( F \right) > v_n^{\left( 3 \right)},m_n^{\left( 2 \right)}\left( F \right) > v_n^{\left( 4 \right)}} \right)} \end{array} $

    其中un(k)=an-1xk+bnvn(k)=cn-1xk+dn.易知

    $ \begin{array}{*{20}{c}} {{{H'}_n}\left( {{x_1},{x_2},{x_3},{x_4},{y_1},{y_2},{y_3},{y_4}} \right) = }\\ {P\left( {N_n^{\left( 1 \right)}\left( {I'} \right) = 0,N_n^{\left( 2 \right)}\left( {I'} \right) \le 1,N_n^{\left( 3 \right)}\left( {J'} \right) = 0,N_n^{\left( 4 \right)}\left( {J'} \right) \le 1} \right.}\\ {\left. {N_n^{\left( 1 \right)}\left( {E'} \right) = 0,N_n^{\left( 2 \right)}\left( {E'} \right) \le 1,N_n^{\left( 3 \right)}\left( {F'} \right) = 0,N_n^{\left( 4 \right)}\left( {F'} \right) \le 1} \right\}} \end{array} $

    其中:I=(0,t1],J=(t1,1],E=(0,t2],F=(t2,1].由文献[1]中的推论5.5.2和引理3,

    $ \begin{array}{*{20}{c}} {\mathop {\lim }\limits_{n \to \infty } {H_n}\left( {{x_1},{x_2},{x_3},{x_4},{y_1},{y_2},{y_3},{y_4}} \right) = }\\ {P\left\{ {N_n^{\left( 1 \right)}\left( {I'} \right) = 0,N_n^{\left( 2 \right)}\left( {I'} \right) \le 1} \right\} \times P\left\{ {N_n^{\left( 3 \right)}\left( {J'} \right) = 0,N_n^{\left( 4 \right)}\left( {J'} \right) \le 1} \right\} \times }\\ {P\left\{ {N_n^{\left( 1 \right)}\left( {E'} \right) = 0,N_n^{\left( 2 \right)}\left( {E'} \right) \le 1} \right\} \times P\left\{ {N_n^{\left( 3 \right)}\left( {F'} \right) = 0,N_n^{\left( 4 \right)}\left( {F'} \right) \le 1} \right\} = }\\ {\left( {\exp \left\{ { - {\theta _1}{t_1}{\tau _2}} \right\}\left( {{t_1}\left( {{\tau _2} - {\tau _1}} \right) + 1} \right)\exp \left\{ {{\theta _1}\left( {1 - {t_1}} \right){\tau _4}} \right\}\left( {\left( {1 - {t_1}} \right)\left( {{\tau _4} - {\tau _3}} \right) + 1} \right)} \right.}\\ {\left. {\exp \left\{ { - {\theta _2}{t_2}{\eta _2}} \right\}\left( {{t_2}\left( {{\eta _2} - {\eta _1}} \right) + 1} \right)\exp \left\{ {{\theta _2}\left( {1 - {t_2}} \right){\eta _4}} \right\}\left( {\left( {1 - {t_2}} \right)\left( {{\eta _4} - {\eta _3}} \right) + 1} \right)} \right) = }\\ {{{H'}_{{t_1}}}\left( {{x_1},{x_2}} \right){H_{1 - {t_1}}}\left( {{x_3},{x_4}} \right){{H'}_{{t_2}}}\left( {{y_1},{y_2}} \right){{H'}_{1 - {t_2}}}\left( {{y_3},{y_4}} \right) = H'\left( {{x_1},{x_2},{x_3},{x_4},{y_1},{y_2},{y_3},{y_4}} \right)} \end{array} $

    其中

    $ {{H'}_{{t_1}}}\left( {{x_1},{x_2}} \right) = {G^{{\theta _1}{t_1}}}\left( {{x_2}} \right)\left( {\log {G^{{t_1}}}\left( {{x_1}} \right) - \log {G^{{t_1}}}\left( {{x_2}} \right) + 1} \right) $

    由于

    $ \begin{array}{*{20}{c}} {P\left\{ {{{\overline {{L_n}} }^{\left( 2 \right)}}/n \le {t_1},{a_n}\left( {M_n^{\left( 2 \right)} - {b_n}} \right) \le {x_2},{{\underline {{L_n}} }^{\left( 2 \right)}}/n > {t_2},a_n^{ - 1}\left( {m_n^{\left( 2 \right)} - {b_n}} \right) > {y_2}} \right\} = }\\ {P\left\{ {M_n^{\left( 2 \right)}\left( I \right) \le u_n^{\left( 2 \right)},M_n^{\left( 2 \right)}\left( I \right) \ge M_n^{\left( 1 \right)}\left( J \right),m_n^{\left( 2 \right)}\left( F \right) < m_n^{\left( 1 \right)}\left( E \right),m_n^{\left( 2 \right)}\left( F \right) > v_n^{\left( 2 \right)}} \right\} + }\\ {P\left\{ {M_n^{\left( 2 \right)}\left( I \right) \le u_n^{\left( 2 \right)},M_n^{\left( 2 \right)}\left( I \right) \ge M_n^{\left( 1 \right)}\left( J \right),m_n^{\left( 1 \right)}\left( E \right) < \\ m_n^{\left( 1 \right)}\left( F \right) < m_n^{\left( 2 \right)}\left( E \right),m_n^{\left( 1 \right)}\left( F \right) > v_n^{\left( 2 \right)}} \right\} + }\\ {P\left\{ {M_n^{\left( 1 \right)}\left( I \right) \le u_n^{\left( 2 \right)},M_n^{\left( 1 \right)}\left( J \right) > M_n^{\left( 1 \right)}\left( I \right) \ge M_n^{\left( 2 \right)}\left( J \right), \\ m_n^{\left( 2 \right)}\left( F \right) < m_n^{\left( 1 \right)}\left( E \right),m_n^{\left( 2 \right)}\left( F \right) > v_n^{\left( 2 \right)}} \right\} + }\\ {P\left\{ {M_n^{\left( 1 \right)}\left( I \right) \le u_n^{\left( 2 \right)},M_n^{\left( 1 \right)}\left( J \right) > M_n^{\left( 1 \right)}\left( I \right) \ge M_n^{\left( 2 \right)}\left( J \right), \\ m_n^{\left( 1 \right)}\left( E \right) < m_n^{\left( 1 \right)}\left( F \right) < m_n^{\left( 2 \right)}\left( E \right),m_n^{\left( 1 \right)}\left( F \right) > v_n^{\left( 2 \right)}} \right\} = }\\ {P\left\{ {X_n^{\left( 2 \right)} \le {x_2},X_n^{\left( 2 \right)} \ge Y_n^{\left( 1 \right)},Y{'}_n^{\left( 2 \right)} < X{'}_n^{\left( 1 \right)},Y{'}_n^{\left( 2 \right)} > {y_2}} \right\} + \\ P\left\{ {X_n^{\left( 2 \right)} \le {x_2},X_n^{\left( 2 \right)} \ge Y_n^{\left( 1 \right)},X{'}_n^{\left( 1 \right)} < Y{'}_n^{\left( 1 \right)} < } \right.}\\ {\left. {X{'}_n^{\left( 2 \right)},Y{'}_n^{\left( 1 \right)} > {y_2}} \right\} + P\left\{ {X_n^{\left( 1 \right)} \le {x_2},Y_n^{\left( 1 \right)} > X_n^{\left( 1 \right)} \ge Y_n^{\left( 2 \right)},Y{'}_n^{\left( 2 \right)} < X{'}_n^{\left( 1 \right)},Y{'}_n^{\left( 2 \right)} > {y_2}} \right\} + }\\ {P\left\{ {X_n^{\left( 1 \right)} \le {x_2},Y_n^{\left( 1 \right)} > X_n^{\left( 1 \right)} \ge Y_n^{\left( 2 \right)},X{'}_n^{\left( 1 \right)} < Y{'}_n^{\left( 1 \right)} < X{'}_n^{\left( 2 \right)},Y{'}_n^{\left( 1 \right)} > {y_2}} \right\}} \end{array} $

    利用点过程的性质(Xn(1)Xn(2)Yn(1)Yn(2)Xn(1)Xn(2)Yn(1)Yn(2))依分布收敛到(X1X2Y1Y2X1X2Y1Y2),其分布函数H绝对连续.因此,

    $ \begin{array}{*{20}{c}} {\mathop {\lim }\limits_{n \to \infty } P\left\{ {{{\overline {{L_n}} }^{\left( 2 \right)}}/n \le {t_1},a_n^{ - 1}\left( {M_n^{\left( 2 \right)} - {b_n}} \right) \le {x_2},{{\underline {{L_n}} }^{\left( 2 \right)}}/n > {t_2},a_n^{ - 1}\left( {m_n^{\left( 2 \right)} - {b_n}} \right) > {y_2}} \right\} = }\\ {P\left\{ {{X_2} \le {x_2},{X_2} \ge {Y_1},{{Y'}_2} < {{X'}_1},{{Y'}_2} > {y_2}} \right\} + \\ P\left\{ {{X_2} \le {x_2},{X_2} \ge {Y_1},{{X'}_1} < {{Y'}_1} < {{X'}_2},{{Y'}_1} > {y_2}} \right\} + }\\ {P\left\{ {{X_1} \le {x_2},{Y_1} > {X_1} \ge {Y_2},{{Y'}_2} < {{X'}_1},{{Y'}_2} > {y_2}} \right\} + }\\ {P\left\{ {{X_1} \le {x_2},{Y_1} > {X_1} \ge {Y_2},{{X'}_1} < {{Y'}_1} < {{X'}_2},{{Y'}_1} > {y_2}} \right\} = }\\ {{t_1}\left( {1 - {t_2}} \right){G^{{\theta _1}}}\left( x \right)\left( {1 - \log G\left( x \right)} \right){{\left( {1 - H\left( y \right)} \right)}^{{\theta _2}}}\left( {1 - \log \left( {1 - H\left( y \right)} \right)} \right.} \end{array} $
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    The Asymptotic Properties of Locations of the Second Maximum and Minimum of Stationary Sequences
    BU Xue-yan, CHEN Shou-quan     
    School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
    Abstract: In this paper, we discuss the asymptotic independence of the normalized second maximum and its location under a long-range dependence condition. Further, we give the asymptotic independence of the joint locations of the second maximum and the joint locations of the second minimum.
    Key words: point process    dependence conditions    location of maxima    asymptotic independence    
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