光滑平稳高斯过程极值几乎处处极限定理的一个推广（英文）

本文研究了连续均方可微的平稳高斯过程的极限性态.通过选择一个不同于Tan(2013)的权重函数,在较弱的条件下得到了连续均方可微平稳高斯过程极值的一个几乎必然仅限定理,推广了Tan(2013)的结论.     

非平稳可微高斯过程的上穿过点过程的渐近分布

{X(t),0≤t≤T}为均方可微非平稳高斯过程。具有渐近中心化的均值m(t)和常数的方差, NT(·)为{X(t),0≤t≤T}上穿过水平uT的点过程,则在一定的条件下匕穿过点过程NT(·)依分布收敛到一Poisson过程．     

平稳强相依高斯过程之上穿点过程的极限定理

设{x(t),t≥0}是一列标准化的具有连续样本轨道的强相依平稳高斯过程,其相关系数函数为r(t).当r(t)满足一定条件时,证明了高斯过程{x(t),t≥0}上穿和ε-上穿水平u形威的点过程的依分布收敛到一Cox-过程.     

d维平稳高斯过程极集的充分条件及维数

本文研究了d维平稳高斯过程极集的性质,给出了d维平稳高斯过程广义极性的充分条件,并通过一个特殊的Cantor型集的构造将极集的维数与容度巧妙地结合起来,得到了d维平稳高斯过程非极集的Hausdorff维数的下确界.     

推广Einstein场方程为随机微分方程

在这篇论文中,我们推广了Einstein场方程成为随机微分方程: 其几何张量和物质张量的分量都被约定为均方连续和均方可微的随机函数。 我们得以建立一些非常深刻的新观点: a.随机Einstein场方程表示随机物质源决定着空-时的随机结构。这一方程的均方解——均方可微的随机度规函数表征着一类随机空-时微分流形。 b.这类随机空-时微分流形可以解释为浸没在R~n空间中的随机超曲面S。在S中任意运动(包括随机运动)的坐标变换下,ds~2是不变量;而且,物理方程也具有协变性,我们称之为随机协变原理。 c.我们解出了这一随机Einstein场方程的一个特殊的均方解(见§4之(18)式)。     

平稳高斯过程的首离时和末离时的渐近性关系（英文）

受近期工作[Theory Probab.Appl.,2021,66(3):337-347]的启发,本文研究了一类弱相依和强相依平稳高斯过程首离时和末离时的渐近性关系,结果表明:如果高斯过程是弱相依的,则首离时和末离时之间是渐近独立的;如果高斯过程是强相依的,则首离时和末离时之间是渐近相依的.     

Gauss白噪声激励下分数阶导数系统的非平稳响应

研究了Gauss(高斯)白噪声激励下具有分数阶导数阻尼的非线性随机动力系统的非平稳响应.应用等价线性化方法将非线性系统转化为等价的线性系统,之后采用随机平均法获得系统响应满足的FPK(Fokker-Planck-Kolmogorov)方程,其中分数阶导数近似为一个周期函数.使用Galerkin方法求解FPK方程进而得到系统的近似非平稳响应.数值结果验证了方法的正确性和有效性.     

Polish空间上的折扣马氏过程量子化策略的渐近优化北大核心CSCD

该文研究了Polish空间上、带折扣因子的连续时间马尔可夫决策过程(CTMDPs)的量子化平稳策略的渐近最优性问题.首先,建立了折扣最优方程(DOE)及其解的存在性和唯一性.其次,在适当的条件下证明了最优确定性平稳策略的存在性.此外,为了对行动空间进行离散化,构造了一列量子化策略,利用有限行动空间的策略来逼近一般(Polish)空间上的折扣CTMDPs最优平稳策略.最后,通过一个例子来说明该文的渐近逼近结果.     

高斯和非高斯平稳时间序列记忆参数的经验似然检验

本文将经验似然方法运用于高斯的和非高斯的平稳时间序列的长记忆性检验.我们从常用的长记忆模型(ARFIMA)出发,建立了记忆参数的经验似然比检验统计量.从理论上证明了所给的经验似然比渐近服从卡方分布,通过数值模拟和实例分析验证了所给的检验方法对于平稳的ARFIMA模型的长记忆参数检验的有效性.     

d维平稳高斯过程重点的不存在性和象集的一致Hausdorff维数

设X^d(t)(t∈R ）是d维可分的平稳高斯过程,在一定条件下,本文得到了X^d(t)象集的一致Hausdorff维数,证明了X^d(t)没有二重点,Polya过程为其特例。     

On some continuity and differentiability properties of paths of Gaussian processes

The following path properties of real separable Gaussian processes ξ with parameter set an arbitrary interval are established. At every fixed point the paths of ξ are continuous, or differentiable, with probability zero or one. If ξ is measurable, then with probability one its paths have essentially the same points of continuity and differentiability. If ξ is measurable and not mean square continuous or differentiable at every point, then with probability one its paths are almost nowhere continuous or differentiable, respectively. If ξ harmonizable or if it is mean square continuous with stationary increments, then its paths are absolutely continuous with probability one if and only if ξ is mean square differentiable; also mean square differentiability of ξ implies path differentiability with probability one at every fixed point. If ξ is mean square differentiable and stationary, then on every interval with probability one its paths are either differentiable everywhere or nondifferentiable on countable dense subsets. Also a class of harmonizable processes is determined for which of the following are true: (i) with probability one paths are either continuous or unbounded on every interval, and (ii) mean square differentiability implies that with probability one on every interval paths are either differentiable everywhere or nondifferentiable on countable dense subsets.     

高斯序列的最大值的几乎必然极限

本文研究了高斯序列{Xn}最大值的几乎必然极限。利用正态比较引理和对数平均,在有关协方差的某些条件下,得到了最大值的一个几乎必然极限定理.     

Central Limit Theorems for the Number of Maxima and an Estimator of the Second Spectral Moment of a Stationary Gaussian Process, with Application to Hydroscience

Let X = (X_t, t 0) be a mean zero stationary Gaussian process with variance one, assumed to satisfy some conditions on its covariance function r. Central limit theorems and asymptotic variance formulas are provided for estimators of the square root of the second spectral moment of the process and for the number of maxima in an interval, with some applications in hydroscience. A consistent estimator of the asymptotic variance is proposed for the number of maxima.     

On asymptotic normality of estimates for correlation functions of stationary Gaussian processes in the space of continuous functions

We establish conditions of the weak convergence of the empirical correlogram of a stationary Gaussian process to some Gaussian process in the space of continuous functions. We prove that such a convergence holds for a broad class of stationary Gaussian processes with square integrable spectral density.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 11, pp. 1485–1497, November, 1995.This work was financially supported by the Ukrainian State Committee on Science and Technology.     

Almost sure convergence for the maxima of strongly dependent stationary Gaussian vector sequences

In this paper, we prove the almost sure limit theorem of the maxima for a kind of strongly dependent stationary Gaussian vector sequences.     

A functional central limit theorem for empirical processes under a strong mixing condition

This paper introduces a functional central limit theorem for empirical processes endowed with real values from a strictly stationary random field that satisfies an interlaced mixing condition. We proceed by using a common technique from Billingsley (Convergence of probability measures, Wiley, New York, 1999), by first obtaining the limit theorem for the case where the random variables of the strictly stationary ???-mixing random field are uniformly distributed on the interval [0, 1]. We then generalize the result to the case where the absolutely continuous marginal distribution function is not longer uniform. In this case we show that the empirical process endowed with values from the ???-mixing stationary random field, due to the strong mixing condition, doesn??t converge in distribution to a Brownian bridge, but to a continuous Gaussian process with mean zero and the covariance given by the limit of the covariance of the empirical process. The argument for the general case holds similarly by the application of a standard variant of a result of Billingsley (1999) for the space D(???, ??).     

Multiple Wiener-itô integral expansions for level-crossing-count functionals

Summary This paper applies the stochastic calculus of multiple Wiener-Itô integral expansions to express the number of crossings of the mean level by a stationary (discrete- or continuous-time) Gaussian process within a fixed time interval [0,T]. The resulting expansions involve a class of hypergeometric functions, for which recursion and differential relations and some asymptotic properties are derived. The representation obtained for level-crossing counts is applied to prove a central limit theorem of Cuzick (1976) for level crossings in continuous time, using a general central limit theorem of Chambers and Slud (1989a) for processes expressed via multiple Wiener-Itô integral expansions in terms of a stationary Gaussian process. Analogous results are given also for discrete-time processes. This approach proves that the limiting variance is strictly positive, without additional assumptions needed by Cuzick.Research supported by Office of Naval Research contracts N00014-86-K-0007 and N00014-89-J-1051     

On the differentiability of a class of stationary gaussian processes

For a stationary Gaussian process either almost all sample paths are almost everywhere differentiable or almost all sample paths are almost nowhere differentiable. In this paper it is shown by means of an example involving a random lacunary trigonometric series that “almost everywhere differentiable” and “almost nowhere differentiable” cannot in general be replaced by “everywhere differentiable” and “nowhere differentiable”, respectively.     

The almost sure limit theorem for the maxima and minima of strongly dependent Gaussian vector sequences

We derive the joint limiting distribution and the almost sure limit theorem for the maxima and minima for a strongly dependent stationary Gaussian vector sequence.     

Functional limits of empirical distributions in crossing theory

We present a functional limit theorem for the empirical level-crossing behaviour of a stationary Gaussian process. This leads to the well-known Slepian model process for a Gaussian process after an upcrossing of a prescribed level as a weak limit in C-space for an empirically defined finite set of functions.We also stress the importance of choosing a suitable topology by giving some natural examples of continuous and non-continuous functionals.