Some Comparisons for Gaussian Processes
Richard A. Vitale
扩展和变种给出了著名的高斯过程的比较原则的基础上按两两距离排序。
Extensions and variants are given for the well-known comparison principle for Gaussian processes based on ordering by pairwise distance.
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Comparison of f-vectors of Random Polytopes to the Gaussian Distribution
根据固定分布，在R^d中随机选择n个独立的点。这些点的凸包是一个随机多面体。在某些情况下，中心极限定理已被证明为f-向量的组成部分的随机多面体构造在这种和类似的方式。在本文中，我们提供了数值证据表明，根据五种不同的分布所产生的随机多面体的f-向量的组件是近似联合高斯大n。
Comparison Inequalities for Order Statistics of Gaussian Arrays
正态比较引理和Slepian不等式是研究高斯过程的重要工具。本文推广了正态比较引理，导出了两个高斯序列的顺序统计量的各种相关的比较不等式，包括Slepian不等式，所得结果可应用于高斯过程的顺序统计量的上确界的研究中.为了说明可能的应用范围，我们分析了自相似高斯过程的顺序统计量的下尾行为，并导出了混合Gumbel极限定理。
In the work a characterization of difference of multivariate Gaussian measures is found on the family of centered Eucledian balls. In particular, it helps to bound corresponding Kolmogorov distance.
We prove a dimension-free tail comparison between the Euclidean norms of sums of independent random vectors uniformly distributed in centred Euclidean spheres and properly rescaled standard Gaussian random vectors.
Comparison theorems for the small ball probabilities of Gaussian processes in weighted $L_2$-norms
本文证明了加权$L_2$ -范数下Green Gauss过程小球概率的比较定理。在相当一般的权假设下，我们发现了许多经典过程的尖锐小球渐近性。
Convex comparison of Gaussian mixtures
受随机微分方程相关凸性研究启发，该文研究高斯分布与高斯混合分布的凸序比较，给出并讨论凸序的内在充要条件，且示例显示两条件紧密相关。
We provide a generalisation of Pinelis' Rademacher-Gaussian tail comparison to complex coefficients. We also establish uniform bounds on the probability that the magnitude of weighted sums of independent random vectors uniform on Euclidean spheres with matrix coefficients exceeds its second moment.
Expected signature of Gaussian processes with strictly regular kernels
我们计算的期望签名的一类高斯过程，这是一个子类的高斯过程与正常的内核，在这个意义上的Alos，Mazet和Nualart。
Exact confidence intervals of the extended Orey index for Gaussian processes
本文利用高斯二次型的浓度不等式和过程的离散观测值，得到了高斯过程Orey指标的精确置信区间.所得结果适用于高斯过程的Orey指标，不一定有平稳增量。
The problem of (pathwise) large deviations for conditionally continuous Gaussian processes is investigated. The theory of large deviations for Gaussian processes is extended to the wider class of random processes -- the conditionally Gaussian processes. The estimates of level crossing probability for such processes are given as an application.
Expansions for Gaussian processes and Parseval frames
本文导出了巴拿赫空间中高斯随机向量的级数展开式与其再生核希尔伯特空间中Parseval标架之间的精确联系。将所得结果应用于路径连续高斯过程，得到了分数阶Ornstein-Uhlenbeck过程的一个新的最优展开式。最后将这一结果推广到具有凸协方差函数的高斯平稳过程。
Representation of self-similar Gaussian processes
我们开发的典型Volterra表示的自相似高斯过程，通过使用相应的平稳高斯过程的Lamperti变换，其中后者承认一个典型的整表示下的假设，纯非决定性。我们应用自相似高斯过程的表示，推导出一个高斯过程的表达式，在法律上是等价的自相似高斯过程的问题。
We derive the asymptotic behavior for an additive functional of two independent self-similar Gaussian processes when their intersection local time exists, using the method of moments.
The Malliavin integration-by-parts formula is a key ingredient to develop stochastic analysis on the Wiener space. In this article we show that a suitable integration-by-parts formula also characterizes a wide class of Gaussian processes, the so-called Gaussian Fredholm processes.
Weak approximation for Gaussian processes from renewal processes
本文研究了从再生成过程构建高斯过程近似的方法，证明了某些进程可用于近似分数布朗运动等，并提供了确保收敛的充分条件。
$L_2$-small ball asymptotics for some demeaned Gaussian processes
该研究探讨Green高斯过程的中心化操作，推导了若干去均值高斯过程的$L_2$-小球渐近性质，丰富了相关过程的渐近理论结果。
Prediction of Gaussian Volterra Processes with Compound Poisson Jumps
该研究针对带复合泊松跳的高斯Volterra过程，推导得到了其预测规律。
Large Deviation Principle for Enhanced Gaussian Processes
研究提升到N步自由幂零群群的高斯过程的大偏差原理。我们将其应用于一大类高斯过程解除几何粗糙路径。在Hoelder-或模拓扑中，增强（分数阶）布朗运动的大偏差原理作为特例出现.
Differentiating sigma-fields for Gaussian and shifted Gaussian processes
我们在（可能漂移的）高斯过程的一般框架下研究了可微分和不可微分σ-场的概念，并刻画了它们在概率测度等价变化下的不变性.作为应用，我们研究了一类与移位分数布朗运动相关的随机导数。最后，我们建立了一个共同可测版本的微分过程的存在条件，我们概述了随机嵌入方程的一般框架。
Partially Gaussian Stationary Stochastic Processes in Discrete Time
本文给出了一个初等例子，对于离散时间中严格平稳非高斯随机过程的每一个固定正整数$k,$，其所有$k$ -边缘都是高斯的.
On estimation of the Orey index for a class of Gaussian processes
Orey提出了平稳增量高斯过程的指数定义，它决定了该过程样本路径的各种性质。本文推广了不具有平稳增量的二阶随机过程的Orey指标的定义，并从样本路径的离散观测值估计了高斯过程的Orey指标。
We introduce a class of self-similar Gaussian processes and provide sufficient and necessary conditions for a member of the class to admit a unique small scale limit in the Skorokhod space. The class includes several well known processes. An example of application to the problem of estimation is given.
Some superconcentration inequalities for extrema of stationary Gaussian Processes
本文研究平稳高斯过程极值的浓度不等式。它提供了非渐近尾部不等式，充分反映了波动率，并因此提高了标准高斯浓度。的参数依赖于超压缩的方法开发的查特吉的超集中方差界。一些统计图表使这一论述更为完整。
Large deviations for perturbed Gaussian processes and logarithmic asymptotic estimates for some exit probabilities
本文研究由满足大偏差原理的高斯过程经扰动得到的非高斯过程族的大偏差，并据此推导出半空间和象限退出概率的对数渐近估计。
A canonical form for Gaussian periodic processes
本文给出了一组高斯过程的一个表示定理，该定理允许构造具有任意正则性的高斯过程，并将其表示为随机三角级数的极限.我们通过Karhunen-Love定理证明了这个集合等距等价于l2。然后，我们证明，任何这样的过程的轨迹路径的规律性，可以检测到只是通过寻找与他通过等距l2序列的减少率。
We deduce bilateral interrelations between Fourier coefficients for lacunar trigonometric series and modulus of their continuity. We obtain also as an application some conditions for continuity and discontinuity for Gaussian periodic stationary random centered processes.
With motivation from K. Dȩbicki and P. Kisowski (2007), in this paper we derive the exact tail asymptotics of $α(t)$-locally stationary Gaussian processes with non-constant variance functions. We show that some certain variance functions lead to qualitatively new results.
Necessary and Sufficient Conditions for Hölder Continuity of Gaussian Processes
高斯过程的连续性是广泛研究的课题，它最终在Talagrand的优化措施，给出了复杂的充要条件的概念。本文研究了高斯过程的Hölder连续性。事实证明，充要条件可以用一种简单的形式来表述，这种形式是著名的柯尔莫哥洛夫-琴佐夫条件的变体。
Weak convergence towards two independent Gaussian processes from a unique Poisson process
我们考虑两个独立的高斯过程，承认一个代表性的确定性内核的随机积分方面的标准维纳过程。本文从一个唯一的泊松过程出发，构造了两类过程，它们的有限维分布按规律收敛于两个独立的高斯过程的有限维分布.作为这一结果的应用，我们得到的过程族，收敛于法律的分数布朗运动和亚分数布朗运动。
Gaussian limit for Pfaffian point processes
本文证明了一类Pfaffian点过程线性统计量的中心极限定理，并应用于Sine_4和Sine_1过程，得到阶梯函数缩放线性统计量的高斯极限。
We establish optimal logarithmic rates of convergence in the strong invariance principle for multivariate cumulative processes in the Smith's sense. Exponential probabilistic inequalities of Komlós-Major-Tusnády type are obtained. Provided examples include applications to stopped sums and birth and death processes.
Linear filtering problem for infinite-dimensional Gaussian processes is studied, the observation process being finite-dimensional. Integral equations for the filter and for covariance of the error are derived. General results are applied to linear SPDEs driven by Gauss-Volterra process observed at finitely many points of the domain.
Parisian Ruin of Self-similar Gaussian Risk Processes
本文给出了自相似高斯风险过程的巴黎破产概率的精确渐近性。此外，我们得到了巴黎破产时间的正态近似，并得到了巴黎破产时间与经典破产时间之间的渐近关系。
Limit properties of exceedances point processes of scaled stationary Gaussian sequences
在较弱的渐近条件下，我们得到了随机尺度弱相依平稳高斯序列的重态点过程的极限分布。在文献中，类似的结果只适用于压缩平稳高斯序列。在本文中，我们包括另外的情况下，随机膨胀平稳高斯序列的威布尔型随机缩放。证明了压缩和膨胀弱相依平稳高斯序列的最大值和最小值是渐近独立的。
A note on ergodic transformations of self-similar Volterra Gaussian processes
本文导出了一类Volterra型自相似高斯过程的遍历变换，即X_t = int^t_0 z_X（t，s）dW_s，t>0，其中z_X是确定核，W是标准布朗运动.
We show that the central limit theorem for linear statistics over determinantal point processes with $J$-Hermitian kernels holds under fairly general conditions. In particular, We establish Gaussian limit for linear statistics over determinantal point processes on union of two copies of $\mathbb{R}^d$ when the correlation kernels are $J$-Hermitian translation-invariant.
In this paper, we study the asymptotic relation between the maximum of acontinuous order statistics process formed by stationary Gaussian processesand the maximum of this process sampled at discrete time points. It is shown that, these two maxima are asymptotically independent when the Gaussian processes are weakly dependent and the discrete points are sufficient sparse, while for other case, these two maxima are asymptotically dependent.
Functional Limit Theorems for Toeplitz Quadratic Functionals of Continuous time Gaussian Stationary Processes
\noindent本文建立了连续时间高斯平稳过程的Toeplitz型二次泛函所产生的归一化随机过程的$C[0,1]$的弱收敛性。得到了中心和非中心泛函的极限定理。
Quasihelix properties of selected Volterra Gaussian processes
本文研究了几种具有温控、幂加权和对数核的高斯Volterra过程的局部准螺旋和广义准螺旋性质，包括温控分数布朗运动和广义分数布朗运动型过程，探讨了参数值对性质的影响。
The aim of this work is to define and perform a study of local times of all Gaussian processes that have an integral representation over a real interval (that maybe infinite). Very rich, this class of Gaussian processes, contains Volterra processes (and thus fractional Brownian motion), multifractional Brownian motions as well as processes, the regularity of which varies along the time. Using the White Noise-based anticipative stochastic calculus with respect to Gaussian processes developed in [Leb17], we first establish a Tanaka formula. This allows us to define both weighted and non-weighted…
We introduce a Skorokhod type integral and prove an Ito formula for a wide class of Gaussian processes which may exhibit stochastic discontinuities. Our Ito formula unifies and extends the classical one for general (i.e., possibly discontinuous) Gaussian martingales in the sense of Ito integration and the one for stochastically continuous Gaussian non-martingales in the Skorokhod sense, which was first derived in Alos et al. (Ann. Probab. 29, 2001).
Moderate deviations for log-like functions of stationary Gaussian processes
建立了高斯过程非线性函数的中偏差原理。非线性函数不必是局部有界的。特别是，允许对数。(Thus，过程的小偏差是相关的。）离散时间和连续时间都被处理。假设相关函数具有可积幂型衰减。
Gaussian estimates for symmetric simple exclusion processes
我们证明高斯尾估计的$n$粒子演变为对称排斥过程$\bb Z^d$，改进\cite{l}中得到的结果。从这一结果出发，我们导出了一维对称简单排它过程的一个非平衡Boltzmann-Gibbs原理，该原理从一个慢变参数的积测度出发.
Local times for multifractional square Gaussian processes
本文考虑由二重Ito-Wiener积分给出的多重分数次过程，它推广了多重分数次Rosenblatt过程.我们证明了这个过程是连续的，并且有一个平方可积〔的〕。
Dirichlet heat kernel estimates for isotropic Levy processes with Gaussian components in Lipschitz open sets
针对含高斯分量的不连续各向同性Levy过程，研究其在Lipschitz开集中的Dirichlet热核估计，给出双侧界及Varopoulos型估计成立的充要条件。
For real symmetric and complex Hermitian Gaussian processes whose values are $d\times d$ matrices, we characterize the conditions under which the probability that at least $k$ eigenvalues collide is positive for $2\le k\le d$, and we obtain the Hausdorff dimension of the set of collision times.
On Small deviations of Gaussian processes using majorizing measures
我们给出了两个周期高斯过程的例子，具有完全相同的顺序，但从根本上不同的小偏差的熵数。我们的构造是基于经典Knopp关于连续无处可微函数存在性的结果，更确切地说是基于Loud函数。我们还得到了一般的小偏差的下界使用优测度方法。我们的例子表明，我们的界限是尖锐的。我们也将其应用到高斯独立序列和超度量高斯过程的一般类。60 F15、60 G50;次级：60 F05
The local structure of $q$-Ornstein-Uhlenbeck processes and $q$-Brownian motions are investigated, for all $q\in(-1,1)$. These are the classical Markov processes corresponding to the noncommutative $q$-Gaussian processes. These processes have discontinuous sample paths, and the local small jumps are characterized by tangent processes. It is shown that for all $q\in(-1,1)$, the tangent processes at inner domain are scaled Cauchy processes possibly with drifts, and the tangent processes at the boundary of the domain are related to the free $1/2$-stable law via Biane's construction.
Excursions away from a regular point for one-dimensional symmetric Levy processes without Gaussian part
研究了一类不含高斯部分的对称Lévy过程偏离正则点的特征测度.证明了killed过程的调和变换具有Feller性质。应用所得结果证明了漂移测度的极值性，并证明了漂移过程和$ h $ -路过程的几个样本路行为。
Survival exponents for some Gaussian processes
该问题是自相似过程在长时间内不超过一个固定水平的概率的幂律渐近性。对某些高斯过程，包括（-T_1，T），T> T_1>>1中的分数布朗运动（FBM）和（0，T），T>>1中的积分FBM，估计了这种渐近性的指数.
Under certain mild conditions, some limit theorems for functionals of two independent Gaussian processes are obtained. The results apply to general Gaussian processes including fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. A new and interesting phenomenon is that, in comparison with the results for fractional Brownian motion, extra randomness appears in the limiting distributions for Gaussian processes with nonstationary increments, say sub-fractional Brownian motion and bi-fractional Brownian. The results are obtained based on the method of mome…
We investigate the Large Deviation behavior in small time of continuous Gaussian processes. We introduce a general procedure allowing to derive Large Deviation Principles in small time starting from the well understood context of Large Deviation Principles with a small parameter, going beyond the self-similar case. Several motivating examples are also treated.
We obtain a Bernstein type Gaussian concentration inequality for martingales. Our inequality improves the Azuma-Hoeffding inequality for moderate deviations $x$. Following the work of McDiarmid (1989), Talagrand (1996) and Boucheron, Lugosi and Massart (2000,2003), we show that our result can be applied to the concentration of random functions, Erdös-Rényi random graph, and maxima of empirical processes. Several interesting Gaussian concentration inequalities have been obtained.
In this paper we provide an upper bound for the conjunction probability of independent Gaussian smooth processes and then we prove that this bound is a good approximation with exponentially smaller error. Our result confirms the heuristic approximation by Euler characteristic method of Worsley and Friston and also implies the exact value of generalized Pickands constant in a special case. Some results for conjunction probability of correlated processes are also discussed.
New results on uniform convergence in probability for expansions of Gaussian random processes using compactly supported wavelets are given. The main result is valid for general classes of nonstationary processes. An application of the obtained results to stationary processes is also presented. It is shown that the convergence rate of the expansions is exponential.
Convergence rate of wavelet expansions of Gaussian random processes
本文刻画了平稳高斯随机过程小波展开的一般类的一致收敛性速度。考虑了依概率收敛。
Characterization of positively correlated squared Gaussian processes
我们解决了埃文斯在1991年提出的一个猜想的特征的正相关平方高斯向量。我们将这种特征从平方高斯向量扩展到永久向量。作为副结果，我们得到了平方高斯过程无穷整除性的几个等价公式。
Multipoint Schur algorithm, II: generalized moment problems, Gaussian processes and prediction
本文利用Krein-Nudelman [1977]关于广义矩问题的经典理论，定义了一类特殊的随机高斯过程。当然，这个类包含平稳高斯过程。我们得到了这个类的过程的谱表示，我们解决了相应的预测问题。单位圆上的正交有理函数导出了一类高斯过程，为上述构造提供了一个例子。
Sharp asymptotics of the functional quantization problem for Gaussian processes
本文给出了希尔伯特空间上高斯测度的L^2-量化误差，特别是高斯过程的L^2-量化误差的尖锐渐近性质。所施加的条件是特征值的规则变化。
Eigenvalue processes of symmetric tridiagonal matrix-valued processes associated with Gaussian beta ensemble
本文研究与高斯β系综相关的对称三对角矩阵值过程，推导其特征值过程满足的随机微分方程，并利用柯西交错论证给出特征值几乎必然不碰撞的充分条件。
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