Francesco Grotto, Umberto Pappalettera
We introduce a Gaussian measure formally preserved by the 2-dimensional Primitive Equations driven by additive Gaussian noise. Under such measure the stochastic equations under consideration are singular: we propose a solution theory based on the techniques developed by Gubinelli and Jara in \cite{GuJa13} for a hyperviscous version of the equations.
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Exponential concentration for zeroes of stationary Gaussian processes
我们证明了对于任何可积协方差的中心平稳高斯过程，其谱测度具有紧支撑或有限指数矩（以及一些额外的正则性），该过程在$[0,T]$中的零点数在其平均值的$ηT$内，直到$T$概率的指数小。
Random walks on primitive lattice points
我们定义了一个随机游动的集合上的原始点的$\mathbb{Z}^d$。我们证明，对于满足温和条件的测量生成的游走，这些游走在很强的意义上是循环的。也就是说，我们证明了相关的马尔可夫链是正常返的，并且存在唯一的平稳测度的随机游动。
Let $f$ be a zero-mean continuous stationary Gaussian process on ${\mathbb R}$ whose spectral measure vanishes in a $δ$-neighborhood of the origin. Then the probability that $f$ stays non-negative on an interval of length $L$ is at most $e^{-cδ^2 L^2}$ with some absolute $c>0$ and the result is sharp without additional assumptions.
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。
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.
We study the stochastic 3D primitive equations of the atmospheric mechanics. We consider them under a bounded and non-degenerate noise, which is statistically periodic in time with period $1$. In such a case we prove that the associated integer-time Markov chain is mixing, which means that there exists a unique stationary measure to which converge the laws all trajectories of this Markov chain.
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。
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.
Extremes of $α(t)$-Locally stationary Gaussian processes with non-constant variances
从K的动机。Danderbicki和P. Kisowski（2007），本文给出了具有非常数方差函数的$α(t)$ -局部平稳高斯过程的精确尾渐近性.我们表明，一些特定的方差函数导致定性的新结果。
Parameter estimation for SDEs related to stationary Gaussian processes
本文研究一般平稳高斯场泛函的部分和的中心极限定理和非中心极限定理。我们应用所得结果研究了与平稳高斯过程有关的随机微分方程的漂移参数估计问题。
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。
Poisson limit theorem for the number of excursions above high and medium levels by Gaussian stationary sequences
本文研究离散时间高斯平稳过程高值和中等值点过程的渐近行为，并证明其可由泊松簇点过程逼近。
Example of a Gaussian self-similar field with stationary rectangular increments that is not a fractional Brownian sheet
本文研究各向异性自相似随机场，特别是分数布朗单。这个高斯场是分数布朗运动的推广。本文证明了矩形增量自相似场的协方差函数的一些性质。利用Lamperti变换，我们得到了相应平稳场的协方差函数的性质。我们提出了一个例子，高斯自相似场与固定的矩形增量，这是不是一个分数布朗单。
Approximation of stationary solutions of Gaussian driven Stochastic Differential Equations
我们研究了与非马尔可夫随机微分方程相关的欧拉格式的经验测度序列：由平稳增量高斯过程驱动的随机微分方程。我们得到的功能收敛到一个平稳的解决方案，该序列的。最后，我们给出了该平稳解的一些具体性质。我们发现，与马尔可夫随机微分方程相比，它的初始随机值和驱动高斯过程总是相关的。然而，在一个整表示的假设下，我们也得到了解决方案的过去是独立于未来的高斯驱动过程的基础创新过程。
This paper studies the problem of equivalence of Gaussian measures induced by Gaussian random fields (GRFs) with stationary increments and proves a sufficient condition for the equivalence in terms of the behavior of the spectral measures at infinity. The main results extend those of Stein (2004), Van Zanten (2007, 2008) and are applicable to a rich family of nonstationary space-time models with possible anisotropy behavior.
Depending on a parameter $h\in (0,1]$, let $\{X_h(\mathbf{t})$, $\mathbf{t}\in\mathcal{M}_h\}$ be a class of centered Gaussian fields indexed by compact manifolds $\mathcal{M}_h$. For locally stationary Gaussian fields $X_h$, we study the asymptotic excursion probabilities of $X_h$ on $\mathcal{M}_h$. Two cases are considered: (i) $h$ is fixed and (ii) $h\rightarrow0$. These results are extended to obtain the limit behaviors of the extremes of locally stationary $χ$-fields on manifolds.
Extremes of alpha(t)-locally Stationary Gaussian Random Fields
本文导出了有限超立方体上α（t）-局部平稳高斯随机场上确界的渐近性质。我们提出了两个应用程序，我们的结果，第一个处理与极端的ggregate multifractional布朗运动，而第二个应用程序建立的确切渐近的上确界的chi过程所产生的multifractional布朗运动。
We describe two classes of Gaussian self-similar random fields: with strictly stationary rectangular increments and with mild stationary rectangular increments. We find explicit spectral and moving average representations for the fields with strictly stationary rectangular increments and characterize fields with mild stationary rectangular increments by the properties of covariance functions of their Lamperti transformations as well as in terms of their spectral densities. We establish that both classes contain not only fractional Brownian sheets and we provide corresponding examples. As a by-…
Lamperti scaling for fractional Gaussian processes with non-stationary increments
本文研究了具有非平稳增量的分数高斯过程的Lamperti缩放，推导了协方差公式和渐进行为，并通过Langevin型过程展示了平稳高斯过程的重建方法。
In this article, we study the stability of solutions to 3D stochastic primitive equations driven by fractional noise. Since the fractional Brownian motion is essentially different from Brownian motion, lots of stochastic analysis tools are not available to study the exponential stability for the stochastic systems. Therefore, apart from the standard method for the case of Brownian motion, we develop a new method to show that 3D stochastic primitive equations driven by fractional noise converge almost surely exponentially to the stationary solutions. This method may be applied to other stochast…
Limit theorems for weighted nonlinear transformations of Gaussian stationary processes with singular spectra
在权函数的一般条件下，导出了具有多重奇异谱的高斯平稳过程的非线性变换的多元权泛函的极限高斯分布.本文的研究动机是它在非线性回归中的潜在应用，以及具有奇异谱的平稳高斯过程的非线性泛函的渐近推断。
Limit Laws for Maxima of Contracted Stationary Gaussian Sequences
本文的主要结果是压缩平稳高斯随机序列最大值的弱极限和强极限。由于随机收缩，我们引入了一个修改的伯曼条件，这是足够的弱收敛的最大值的缩放样本。在更强的假设下，弱收敛被加强为几乎收敛。
The General Stationary Gaussian Markov Process
我们发现类，${\cal{C}}_k, k \ge 0$，所有零均值平稳高斯过程，$Y(t), ~t \in \reals$与$k$衍生物，其中\begin{equation} Z(t) \equiv (Y^{(0)}(t), Y^{(1)}(t), \ldots, Y^{(k)}(t) ), ~ t \ge 0 \end{equation} \noindent是一个$(k+1)$ -向量马尔可夫过程。(here、$Y^{(0)}(t) = Y(t)$）。
We study the persistence probability of a centered stationary Gaussian process on $\mathbb{Z}$ or $\mathbb{R}$, that is, its probability to remain positive for a long time. We describe the delicate interplay between this probability and the behavior of the spectral measure of the process near zero and infinity.
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。
Assuming certain conditions on the spectral measures of centered stationary Gaussian processes on $\mathbb{R}$ (or ${\mathbb{R}}^2$), we show that the probability of the event that their zero count in an interval (resp., nodal length in a square domain) is larger than $n$, where $n$ is much larger than the expected value of the zero count in that interval (resp., nodal length in that square domain), is exponentially small in $n$.
Gärtner-Ellis condition for squared asymptotically stationary Gaussian processes
建立了渐近平稳高斯过程平方的Gärtner-Ellis条件。对于给定任何固定的初始点的条件分布，也有同样的极限，这就产生了弱乘法遍历性。极限被证明是拉普拉斯变换的卷积的Γ分布与泊松复合的指数。一个基于Wiener-Hopf分解的证明导致了一个回归问题的极限的概率解释。
Employing the optimal fluctuation method (OFM), we study the large deviation function of long-time averages $(1/T)\int_{-T/2}^{T/2} x^n(t) dt$, $n=1,2, \dots$, of centered stationary Gaussian processes. These processes are correlated and, in general, non-Markovian. We show that the anomalous scaling with time of the large-deviation function, recently observed for $n>2$ for the particular case of the Ornstein-Uhlenbeck process, holds for a whole class of stationary Gaussian processes.
Persistence probabilities of two-sided (integrated) sums of correlated stationary Gaussian sequences
本文研究了一类双边离散高斯序列的持续生存概率，它们分别是分数布朗运动和积分分数布朗运动的离散模拟.我们的结果推广了[Molchan，Commun.数学物理，1999]和[Molchen，J. Stat.物理、[2017]一个广泛的类离散时间过程。
Parameter Estimation for the Langevin Equation with Stationary-Increment Gaussian Noise
研究了带平稳增量高斯噪声的朗之万方程。本文证明了均值回归参数的替代估计的强相合性和具有Berry-Esseen界的渐近正态性。的条件和结果表示的方差函数的噪声。我们考虑连续和离散观测的情况。作为例子，我们考虑分数和bifractial Ornstein-Uhlenbeck过程。最后，我们讨论了极大似然估计和最小二乘估计。
A criterion for testing hypotheses about the covariance function of a stationary Gaussian stochastic process
考虑可测平稳高斯随机过程。本文构造了一个准则，利用$L_p(\mathbb {T}),\,p\geq1$空间中的范数估计来检验关于这类过程的协方差函数的假设。
Stochastic integration for a wide class of Gaussian stationary increment processes using an extension of the S-transform
给定一个具有谱密度的高斯平稳增量过程，利用Hida的白噪声空间理论，我们证明了关于这个过程的Wick-Ito积分可以自然地得到.我们使用Bochner-Minlos定理将过程关联到一个概率空间，并在该空间中定义S变换的对应物。然后，我们使用这个变换来定义随机积分，并证明了相关的伊藤公式。
Spectral representation of correlation functions for zeros of Gaussian power series with stationary coefficients
本文研究系数为离散平稳高斯过程的高斯解析函数，利用谱测度重新考察极限定理、数值检验例子，并给出零点集n点相关函数的积分表示。
On the asymptotic behavior of first passage time densities for stationary Gaussian processes
利用类Rice级数展开，确定了一类平稳高斯过程首达时间密度函数通过某些时变边界（包括周期边界）的渐近性态.当边界是渐近常数或渐近周期时，给出了密度渐近呈现指数行为的充分条件。
Tests of independence for pairs of paths of non-stationary Gaussian processes
本文为非平稳高斯过程（标准布朗运动与分数布朗运动）路径对的独立性检验提供了理论结果。
In this paper we study the local times of vector-valued Gaussian fields that are `diagonally operator-self-similar' and whose increments are stationary. Denoting the local time of such a Gaussian field around the spatial origin and over the temporal unit hypercube by $Z$, we show that there exists $λ\in(0,1)$ such that under some quite weak conditions, $\lim_{n\rightarrow +\infty}\frac{\sqrt[n]{\mathbb{E}(Z^n)}}{n^λ}$ and $\lim_{x\rightarrow +\infty}\frac{-\log \mathbb{P}(Z>x)}{x^{\frac{1}λ}}$ both exist and are strictly positive (possibly $+\infty$). Moreover, we show that if the underlying G…
Given that a stationary Gaussian process is above a high threshold, the length of time it spends before going below that threshold is studied. The asymptotic order is determined by the smoothness of the sample paths, which in turn is a function of the tails of the spectral measure. Two disjoint regimes are studied - one in which the second spectral moment is finite and the other in which the tails of the spectral measure are regularly varying and the second moment is infinite.
Persistence probabilities in centered, stationary, Gaussian processes in discrete time
在离散时间平稳高斯过程的谱测度满足一定条件的情况下，得到了该过程持续生存概率的下界。给出的例子表明，持久性概率可以比指数衰减更快。它表明，如果谱测度是不是奇异的，那么在持久性概率的指数不能比二次增长更快。一个例子，出现（从数值证据），以实现这一下界。
Exact asymptotics of supremum of a stationary Gaussian process over a random interval
设$\{X(t) : t \in [0, \infty) \}$为中心平稳高斯过程。我们研究了$\pr (\sup_{s \in [0,T]} X(t) > u)$的精确渐近性，即$u \to \infty$，其中$T$是与\{X（t）\}非负随机变量无关的.看来，$T$的沉重影响的形式的渐近，导致三种情况下：可积的$T$的情况下，$T$的情况下，有规律地变化的尾部分布参数$λ\in(0,1)$和$T$的情况下，有缓慢变化的尾部分布。
We prove convergence of Hausdorff measure of level sets of smooth Gaussian fields when the levels converge. Given two coupled stationary fields $f_1, f_2$ , we estimate the difference of Hausdorff measure of level sets in expectation, in terms of $C^2$-fluctuations of the field $F=f_1-f_2$. The main idea in the proof is to represent difference in volume as an integral of mean curvature using the divergence theorem. This approach is different from using Kac-Rice type formula as main tool in the analysis.
Small deviations of sums of correlated stationary Gaussian sequences
研究了平稳高斯序列和的小偏差概率。对于常边界和边界趋于零的情形，我们得到了相当一般的结果。对于边界趋于无穷大的情形，我们把注意力集中在分数布朗运动（FBM）的离散模拟上。事实证明，SDP的下界可以转移到离散时间设置下，通常的假设，意味着弱收敛，而相应的上界的转移必然需要更深入的知识的频谱结构的基础平稳序列。
Small Deviations of Smooth Stationary Gaussian Processes
研究了一类在贝叶斯统计推断中起重要作用的平稳高斯过程的小偏差概率。我们的计算是基于适当的修改的熵方法由于Kuelbs，李，林德以及经典的结果熵类的解析函数。他们还涉及Tsirelson的上界的小偏差，并揭示了一些关于该估计的锐度的限制。
Let $\{X_i(t):\, t\in S\subset \R^d \}_{i=1,2,\ldots,n}$ be independent copies of a stationary centered Gaussian field with almost surely smooth sample paths. In this paper, we are interested in the conjunction probability defined as $\PP \left( \exists t\in S: \, X_i(t) \geq u, \, \forall i=1,2,\ldots,n \right)$ for a given threshold level $u$. As $u\rightarrow \infty$, we will provide an asymptotic formula for the conjunction probability. This asymptotic formula is derived from the behaviour of the volume of the set of local maximum points. The proof relies on a result of Aza\"ıs and Wschebo…
Persistence and entropic repulsion of stationary Gaussian fields with spectral singularity at the origin
本文研究了具有原点谱奇异性的平稳高斯场的持久概率的精确对数渐近行为，并确定了在持久条件下熵排斥分布，该研究适用于一般d维平稳高斯场，结果依赖于参数α和d，并通过α-里斯核的容量和平衡势显式表达。
On the extreme order statistics for stationary Gaussian sequences subject to random missing observations
本文研究了存在随机缺失观测的平稳高斯序列，通过分析多维超限点过程的极限性质，导出了原序列与观测序列的极值顺序统计量及最大值位置和高度的联合极限分布。
On the peak height distribution of non-stationary Gaussian random fields: 1D general covariance and scale space
本文推导一维一般协方差非平稳高斯过程的显式峰值高度分布，并推广至多维尺度空间场，提出两种高效数值算法。
Limit Laws for Extremes of Dependent Stationary Gaussian Arrays
本文证明了弱相依二元平稳高斯三角阵的分量极大值在归一化为Hüsler-Reiss分布后收敛于分布。在强相依假设下，证明了极大值的极限分布是一个二元高斯分布和Hüsler-Reiss分布的混合分布.我们的论文的另一个发现是，即使在Hüsler和Reiss（1989）的设置中，分量最大值和分量最小值仍然渐近独立，进一步允许弱相关性。在Berman条件下，我们进一步得到了三角阵各分量的几乎必然极限定理。
Maximum likelihood drift estimation for Gaussian process with stationary increments
本文讨论了回归模型$X_t = θt + B_t$，$t\in[0, T ]$，其中$B=\{B_t, t\geq 0\}$是一个平稳增量的中心高斯过程.本文研究了未知参数$θ$的估计问题，并建立了用解积分方程表示的似然函数的公式。然后我们找到极大似然估计并证明了它的强相合性。所得结果推广了分数阶和混合分数阶布朗运动的已知结果.
Piterbarg's max-discretisation theorem for stationary vector Gaussian processes observed on different grids
在本文中，我们推导出Piterbarg的最大离散定理，考虑两个不同的网格中心平稳向量高斯过程。到目前为止，在这个方向上的文献结果已经得出了联合分布的最大高斯过程在$[0,T]$和网格$ \mathfrak{R}(δ_1(T))=\{kδ_1(T): k=0,1,\cdots\}$。在本文中，我们扩展了最近的研究结果，考虑另外的\bE{maximum}在另一个网格$ \mathfrak{R}(δ_2(T))$。我们通过令$T\to \infty$得到了在不同网格选择下平稳高斯向量过程最大值的联合极限分布。
We investigate asymptotics of the tail distribution of sojourn time $$ \int_0^T \mathbb{I}(X(t)> u)dt, $$ as $u\to\infty$, where $X$ is a centered stationary Gaussian process and $T$ is an independent of $X$ nonnegative random variable. The heaviness of the tail distribution of $T$ impacts the form of the asymptotics, leading to four scenarios: the case of integrable $T$, the case of regularly varying $T$ with index $λ=1$ and index $λ\in(0,1)$ and the case of slowly varying tail distribution of $T$. The derived findings are illustrated by the analysis of the class of fractional Ornstein-Uhlenb…
On Piterbarg Max-discretisation Theorem for Multivariate Stationary Gaussian Processes
设$\{X(t), t\geq0\}$是均值为零、方差为单位的平稳高斯过程。在Piterbarg（2004）中导出的一个深入结果，我们称之为Piterbarg的最大离散化定理，给出了连续时间最大值$M(T)=\max_{t\in [0,T]} X(t), $和最大值$M^δ(T)=\max_{t\in \mathfrak{R}(δ)}X(t), $的联合渐近行为（$T\to \infty$），其中$\mathfrak{R}(δ) \subset [0,T]$是距离$δ=δ(T)$的均匀网格。在相关函数的某些渐近限制下，Piterbarg的最大离散化定理表明，对于极限结果，重要的是要知道$δ(T)$接近0的速度为$T\to \infty$。本文导出了多元平稳高斯过程的上述定理。
Stationary systems of Gaussian processes
我们在$\mathbb{R}$上描述了所有具有独立性、高斯性和平稳性的可数粒子系统。更准确地说，我们认为粒子在实数线上开始的泊松点过程的点与强度测量$\mathfrak{m}$和移动相互独立的一些高斯过程$ξ$的法律。我们分类所有对$(\mathfrak{m},ξ)$产生一个固定的粒子系统，获得三个家庭的例子。在第一个平凡族中，测度$\mathfrak{m}$是任意的，而过程$ξ$是平稳的。在第二族中，测度$\mathfrak{m}$是勒贝格测度的倍数，并且$ξ$本质上是具有线性漂移的高斯平稳增量过程。在第三个最有趣的族中，测度$\mathfrak{m}$的密度形式为$αe^{-λx}$，其中$α>0$，$λ\in\mathbb{R}$，而过程$ξ$的密度形式为$ξ(t)=W(t)-λσ^2(t)/2+c$，其中$W$是具有平稳增量的零均值高斯过程，$σ^2(t)=\operatorname {Var}W(t)$和$c\in\mathbb{R}$。
The winding of stationary Gaussian processes
本文研究了连续可微高斯平稳过程$f:\mathbb{R}\to\mathbb{C}$在区间$[0,T]$中的缠绕.我们给出了这个随机变量的均值和方差的公式。方差总是增长至少与$T$线性，并给出了它是渐近线性或二次的条件。此外，我们表明，如果协方差函数连同其二阶导数是在$L^2(\mathbb{R})$，那么绕组服从中心极限定理。这些结果对应于Cuzick，Slud等人关于实值平稳高斯函数零点的类似结果。
Long gaps between sign-changes of Gaussian Stationary Processes
研究了大区间$[0,N]$上实值平稳过程为正的概率。我们证明了如果在原点的某个邻域中，过程的谱测度具有远离零和无穷大的密度，则该概率的衰减在$N$中的两个指数函数之间有界.这推广了特殊情况下获得的类似界限，例如Artezana，Buckley，Marzo，Olsen最近的结果。
Let $\{Z(τ,s), (τ,s)\in [a,b]\times[0,T]\}$ with some positive constants $a,b,T$ be a centered Gaussian random field with variance function $σ^{2}(τ,s)$ satisfying $σ^{2}(τ,s)=σ^{2}(τ)$. We firstly derive the exact tail asymptotics for the maximum $M_{H}(T)=\max_{(τ,s)\in[a,b]\times[0,T]}Z(τ,s)/σ(τ)$ up crossing some level $u$ with any fixed $0<a<b<\infty$ and $T>0$; and we further derive the extreme limit law for $M_{H}(T)$. As applications of the main results, we derive the exact tail asymptotics and the extreme limit law for Shepp statistics with stationary Gaussian process, fractional Brow…
The density of zeros of random power series with stationary complex Gaussian coefficients
研究具有平稳复高斯系数的随机幂级数，分析了收敛圆边界附近零点径向密度的渐近行为，揭示了系数依赖性降低零点密度的机制及谱测度支撑与解析延拓的关系。
We consider the winding number of planar stationary Gaussian processes defined on the line. Under mild conditions, we obtain the asymptotic variance and the Central Limit Theorem for the winding number as the time horizon tends to infinity. In the asymptotic regime, our discrete approach is equivalent to the continuous one studied previously in the literature and our main result extends the existing ones. Our model allows for a general dependence of the coordinates of the process and non-differentiability of one of them. Furthermore, beyond our general framework, we consider as examples an app…
Second-order continuous-time non-stationary Gaussian autoregression
研究了高斯白噪声驱动下二阶线性随机常微分方程中未知参数的极大似然估计的渐近性态.重点是在非遍历的情况下，当相应的特征方程的根不都在左半平面。
We study the variance of the number of zeroes of a stationary Gaussian process on a long interval. We give a simple asymptotic description under mild mixing conditions. This allows us to characterise minimal and maximal growth. We show that a small (symmetrised) atom in the spectral measure at a special frequency does not affect the asymptotic growth of the variance, while an atom at any other frequency results in maximal growth. Our results allow us to analyse a large number of interesting examples.
Gaussian Processes Generated By Monotonically Modulated Stationary Kernels
该研究聚焦单调调制平稳核生成的高斯过程，构建了原空间与调制空间的显式等距映射，还推导得到区间内期望零点数的精确表达式，为相关高斯过程分析提供了理论支撑。
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