一类拟平稳序列极值的极限律

Ｋ．Ｆ．Ｔｕｒｋｍａｎ讨论了一类拟平稳序列最大值的渐近分布。本文利用点过程收全党一理得到水平超出点过程的收敛定理和第ｒ个最大值的渐近分布及前ｒ个最大值的联合渐近分布。     

一类向量高斯过程之上穿过点过程的渐近分布

｛Ｘ（ｔ），ｔ≥｝为ｐ维高斯过程，在一定条件下，本文得到了｛Ｘ（ｔ），０≤ｔ≤Ｔ｝对水平ＵＴ（＞０）的ε－上穿过次数所形成的点过程的渐近分布（Ｔ→∞）。证明了Ｐ个分量点过程的渐近独立性。     

一类马尔可夫序列最大值的极限律

主要研究一类马尔可夫序列{Xn,n≥0}的最大值的极限分布.导出了这类序列最大值和最小值的分布表达式,利用经典极值理论,建立了规范化最大值max{X0,X1,…,Xn}与i.i.d序列{ξn,n≥1}的规范化最大值max{1ξ,2ξ,…,ξn+1}具有相同极限律的条件.     

一类非平稳高斯序列超过数点过程与和的渐近性

设{X_i}_(i=1)~∞是标准化非平稳高斯序列,N_n为X_1,X_2,…,X_n依次对水平μ_(n1),μ_(n2),…,μ_(nn)的超过数形成的点过程.记Υ_(ij)=X_iX_j,S_n=■X_i.当Υ_(ij)满足一定条件时,证明了N_n依分布收敛到Poisson过程,且N_n与S_n渐近独立.     

平稳正态序列超过数点过程与部分和的渐近联合分布

{Xi}为平稳正态序列,具有EX1=0,EX12=1,ρn=EX1Xn 1.对于水平un= ,记在 的条件下,得到了Nn(B)与Sn的渐近联合分布,同时也给出了极值与Sn的渐近联合分布.     

含趋势项强相依非平稳序列的两个重要分布

{ηn}为平稳标准化正态序列,相关系数r|t-j|=Cov(ηu,ηj),若rnlogn→∞时,Leadbetter[1]等得到了序列最大值的渐近分布.本文考虑非平稳带有趋势项序列{ηn},得到了序列最大值的渐近分布和最大值与部分和的联合渐近分布.     

一类相依正态序列超过数点过程的渐近分布

{X_n,n≥1}为存在样本缺失的标准化平稳正态序列,相关系数r_n=EX_1X_(1+n).(?)_n与(?)_n分别为观测到与未观测到的子样形成的超过数点过程.令N_n=(?)_n+(?)_n.本文研究r_nln→ρ∈[0,∞)时超过数点过程N_n,(?)_n与(?)_n的弱收敛性及顺序统计量的联合渐近分布.     

非平稳高斯序列超过数点过程与部分和的联合渐近分布

设$\{X_{i}\}^{\infty}_{i=1}$是标准化非平稳高斯序列, $N_{n}$为$X_{1},X_{2},\cdots,X_{n}$对水平$\mu_{n}(x)$的超过数形成的点过程, $r_{ij}=\ep X_{i}X_{j}$, $S_{n}=\tsm_{i=1}^{n}X_{i}$. 在$r_{ij}$满足一定条件时, 本文得到了$N_{n}$与$S_{n}$的渐近独立性.     

一类广泛超稳定过程占位时的渐近行为

本文在「１」的基础上，针对非常值分枝率对应的一类广泛超α－稳定过程占位时过程，证明了在非临界情况下其渐近行为与「１」中相同，但在临界时其渐近性依赖于分枝率在无穷远点的极限行为，从而得到了更为精细的结果。     

Extremes of vector-valued Gaussian processes

The seminal papers of Pickands (Pickands, 1967; Pickands, 1969) paved the way for a systematic study of high exceedance probabilities of both stationary and non-stationary Gaussian processes. Yet, in the vector-valued setting, due to the lack of key tools including Slepian’s Lemma, there has not been any methodological development in the literature for the study of extremes of vector-valued Gaussian processes. In this contribution we develop the uniform double-sum method for the vector-valued setting, obtaining the exact asymptotics of the high exceedance probabilities for both stationary and n on-stationary Gaussian processes. We apply our findings to the operator fractional Brownian motion and Ornstein–Uhlenbeck process.     

Extended Classical Asymptotic Expansions in the Case of Gaussian Limit Distribution

Let X, X ₁, X ₂,... be a sequence of independent and identically distributed random variables with common distribution function F. Denote by F _n the distribution function of centered and normed sum S _n . Let F belong to the domain of attraction of the standard normal law , that is, lim F _n (x)= (x), as n , uniformly in x . We obtain extended asymptotic expansions for the particular case where the distribution function F has the density p(x) = cx ^––1 ln(x), x > r, where 2, , c > 0, and r > 1. We write the classical asymptotic expansion (in powers of n ^–1/2) and then add new terms of orders n ^–/2 ln n, n ^–/2 ln^-1 n, etc., where 0.     

Central Limit Theorems for Level Functionals of Stationary Gaussian Processes and Fields

We introduce a general method, which combines the one developed by authors in 1997 and one derived from the work of Malevich,⁽¹⁷⁾ Cuzick⁽⁷⁾ and mainly Berman,⁽³⁾ to provide in an easy way a CLT for level functionals of a Gaussian process, as well as a CLT for the length of a level curve of a Gaussian field.     

Asymptotic Expansions for the Distribution of the Maximum of Gaussian Random Fields

Some asymptotic results are proved for the distribution of the maximum of a centered Gaussian random field with unit variance on a compact subset S of ^N . They are obtained by a Rice method and the evaluation of some moments of the number of local maxima of the Gaussian field above an high level inside S and on the border S. Depending on the geometry of the border we give up to N+1 terms of the expansion sometimes with exponentially small remainder. Application to waves maximum is shown.     

Degree Sequences Beyond Power Laws in Complex Networks

Many complex networks possess vertex-degree distributions in a power-law form of $ck^{-\gamma}$, where $k$ is the degree variable and $c$ and $\gamma$ are constants. To better understand the mechanism of power-law formation in real-world networks, it is effective to analyze their degree variable sequences. We had shown before that, for a scale-free network of size $N$ ,if its vertex-degree sequence is $k_11$ , then the length $l$ of the vertex-degree sequence is of order $logN$ . In the present paper, we further study complex networks with more general distributions and prove that the same conclusion holds even for non-network type of complex systems. In addition, we support the conclusion by verifying many real-world network and system examples. We finally discuss some potential applications of the new finding in various fields of science, technology and society.     

一类适应随机变量序列的强极限定理

研究了一类适应随机变量序列的局部收敛性,推广了文献[1]中的结论.并在假定部分和序列为极限鞅时,得到了极限鞅的强极限定理.最后给出了*-mixing序列的强大数定律.     

Asymptotic Poisson Character of Extremes in Non-Stationary Gaussian Models

Let X be a non-stationary Gaussian process, asymptotically centered with constant variance. Let u be a positive real. Define R_u(t) as the number of upcrossings of level u by the process X on the interval (0, t]. Under some conditions we prove that the sequence of point processes (R_u)_u>0 converges weakly, after normalization, to a standard Poisson process as u tends to infinity. In consequence of this study we obtain the weak convergence of the normalized supremum to a Gumbel distribution.AMS 2000 Subject Classifications Primary—60G70, 60G15     

极限方程有奇性的一类六阶椭圆型方程混合边值问题解的渐近式

该文研究极限方程在部分边界上为退缩椭圆型(椭圆-抛物)的一类六阶椭圆型方程混合边值问题的奇摄动,在适当的假设下,应用改进了的多重尺度法,求得其解包括边界层和套层在内除了半圆域的两个角点外,在整个半圆域中有任意阶的一致有效的渐近展开式.