A. I. Nazarov, Ya. Yu. Nikitin
We introduce a new approach to the spectral equivalence of Gaussian processes and fields, based on the methods of operator theory in Hilbert space. Besides several new results including identities in law of quadratic norms for integrated and multiply integrated Gaussian random functions we give an application to goodness-of-fit testing.
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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.
Asymptotic equivalence for density estimation and gaussian white noise: An extension
本文的目的是提出一个著名的渐近等价性之间的密度估计实验和高斯白噪声模型的扩展。我们的扩展包括扩大非参数类的容许密度。更确切地说，我们提出了一种方法，允许密度定义在R的任何子区间，也考虑了一些不连续或无界密度（只要不连续和无界模式是某种先验已知的）。我们将采用的等效概念是统计模型之间的Le Cam距离。结果是建设性的：所有的渐近等价建立通过构造显式马尔可夫核。
Fredholm representation of multiparameter Gaussian processes with applications to equivalence in law and series expansions
证明了每一个方差函数可积的多参数高斯过程都存在关于布朗单的Fredholm型Wiener整表示。该表示中的Fredholm核可以构造为协方差的唯一对称平方根。利用Fredholm表示分析了多参数高斯过程的等价性，并给出了如何利用Fredholm核构造多参数高斯过程的级数展开式.
On the equivalence of probability spaces
对于一般测度空间$(M,\mathscr F, σ)$的sigma-代数$\mathscr F$所索引的一类高斯过程$W$，给出了这类高斯过程的二次变差表示有效的充要条件，从而将$A\in\mathscr F$的$σ(A)$恢复为$W$在$A$上的二次变差.我们还提供了一个谐波分析表示这类一般的过程。我们将这两个结果：$(i)$计算广义伊藤积分和$(ii)$证明了一个明确的，和测量理论的等价公式，实现了两种方法之间的等价高斯过程，其中一个选择的样本空间是传统的路径空间，和其他地方，它是施瓦茨空间的回火分布。
Given an instance $h$ of the Gaussian free field on a planar domain $D$ and a constant $γ\in (0,2)$, one can use various regularization procedures to make sense of the Liouville quantum gravity area measure $μ:= e^{γh(z)} dz.$ It is known that the field $h$ a.s. determines the measure $μ_h$. We show that the converse is true: namely, $h$ is measurably determined by $μ_h$. More generally, given a random closed fractal subset $\mathcal A$ endowed with a Frostman measure $σ$ whose support is $\mathcal A$ (independent of $h$), a Gaussian multiplicative chaos measure $μ_{σ,h}$ can be constructed. W…
Gaussian Equivalence for Multi-Head Self-Attention
本文利用随机矩阵理论建立多头自注意力的高斯等价性，证明用重缩放分数加噪声替换softmax可保持谱律，并分离头分配与投影宽度的影响。
Asymptotic equivalence for pure jump Lévy processes with unknown Lévy density and Gaussian white noise
本文的目的是建立一个全局渐近等价的实验产生的离散（高频）或连续观察的L{é}vy过程的路径和高斯白噪声实验观察到的时间T，T趋于$\infty$。这些近似是在Le Cam距离的意义下，在未知L{é}vy密度的某些光滑条件下给出的。所有的渐近等价性都是通过构造显式马尔可夫核来建立的，这些显式马尔可夫核可以用来从另一个实验中重现一个实验。
Gaussian Processes and Reproducing Kernel Hilbert Spaces: Connections and Equivalences
本专著研究机器学习等领域中基于正定核的高斯过程概率方法与再生核希尔伯特空间非概率方法的联系与等价性，建立统一视角以衔接相关研究。
We prove the equivalence of the local property for an irreducible regular Dirichlet form and the Markov property for the Gaussian field associated with the Dirichlet form. Moreover we introduce a strong Markov property for Gaussian fields and present some sufficient conditions for this to hold.
Spectral Purification in Reversible Markov Chains: Hidden Parameters, Observable Equivalence, and Finite-Time Rigidity
本文研究了可逆马尔可夫链中有限时间谱刚性的本质，通过精确的谱弛豫动力学揭示了谱刚性的时间结构，并建立了谱分离比控制谱刚性出现的理论框架，同时提出了基于谱熵理论的精确误差身份和自适应停止准则。
Topological equivalence for discontinuous random dynamical systems and applications
在定义了双边时间的非高斯Lévy过程之后，考虑了具有这类Lévy过程的随机微分方程.这些随机微分方程的解路径在时间上具有可数个跳跃间断。建立了这类伊藤随机微分方程与其变换后的随机微分方程的拓扑等价（或共轭）性.从而证明了伊藤随机微分方程线性化的随机Hartman-Grobman定理。进一步，对Marcus随机微分方程，利用这种拓扑等价证明了全局随机吸引子的存在性.
Gaussian Equivalence for Self-Attention: Asymptotic Spectral Analysis of Attention Matrix
本文研究了注意力矩阵的奇异值谱，首次证明了注意力的高斯等价性，揭示了线性化阈值及为何非逐元素操作的注意力在特定条件下可严格满足高斯等价性。
Sample paths properties of Gaussian fields with equivalent spectral densities
证明了如果$X$和$Y$是两个谱密度相等的高斯场，则它们在任意连续嵌入$\mathcal{C}^0(K)$的可分巴拿赫空间中具有相同的样本路径性质，其中$K$是$\mathbb{R}^d$的紧集.
We obtain sufficient conditions of stochastic equivalence of Gaussian random fields with special covariance function. These results generalize Doob's transformation (condition of stochastic equivalence of a Gaussian and a Wiener processes) to the case of random fields (condition of stochastic equivalence of a Gaussian process and a Brownian sheet). We look at the problem of finding the distribution of supremum of a Brownian sheet on a set with a dimension lower than the dimension of the field. We consider the probability of a Brownian sheet with a certain drift to attain zero level. The obtain…
Stochastic Analysis of Gaussian Processes via Fredholm Representation
证明了每一个方差函数可积的可分高斯过程都存在关于布朗运动的Fredholm表示。本文将Fredholm表示推广为一个转移原理，并利用它发展了随机分析，给出了Fredholm表示在等价律、桥、级数展开、随机微分方程和极大似然估计等方面的应用.
Local spectral statistics of Gaussian matrices with correlated entries
证明了一类具有相关元素的平移不变高斯随机矩阵系综的最优局部律、体普适性和预解式非对角元的非平凡衰减性。
The $n$th order fractional Brownian motion was introduced by Perrin et al. It is the (upto a multiplicative constant) unique self-similar Gaussian process with Hurst index $H \in (n-1,n)$, having $n$th order stationary increments. We provide a transfer principle for the $n$th order fractional Brownian motion, i.e., we construct a Brownian motion from the $n$the order fractional Brownian motion and then represent the $n$the order fractional Brownian motion by using the Brownian motion in a non-anticipative way so that the filtrations of the $n$the order fractional Brownian motion and the associ…
We give a non-asymptotic bound on the spectral norm of a $d\times d$ matrix $X$ with centered jointly Gaussian entries in terms of the covariance matrix of the entries. In some cases, this estimate is sharp and removes the $\sqrt{\log d}$ factor in the noncommutative Khintchine inequality.
As an important topic in Mathematical Physics and statistics, random matrices theory has found uses in many aspects of modern physics and multivariate analysis. This paper is to investigate the Gaussian fluctuations for linear spectral statistics (LSS) of Wigner beta ensembles. We first establish a central limit theorem (CLT) for LSS of Wigner quaternion matrices, then give a general CLT for Wigner $β$ ensembles.
Asymptotic statistical equivalence for ergodic diffusions: the multidimensional case
在Le Cam意义下的渐近局部等价性是建立在多维遍历扩散和伴随的高斯位移实验序列的漂移上的推论。非参数局部邻域可以达到任何尺寸，提供的漂移的规律性是足够大的。此外，异方差高斯回归实验，这也是局部渐近等价的，不依赖于本地化的中心。对于等价的一个方向，构造了一个显式的马尔可夫核。
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。
Equivalence of measures and asymptotically optimal linear prediction for Gaussian random fields with fractional-order covariance operators
本文研究分数阶协方差高斯测度的等价性及错误设定下线性预测的一致渐近最优性，并以广义 Whittle–Matérn 场解释平稳与非平稳模型预测性能相近的原因。
Gaussian upper heat kernel bounds and Faber-Krahn inequalities on graphs
研究图上相对Faber-Krahn不等式与高斯上热核界及体积加倍条件的等价性，通过引入局部正则性和可变维数推广，并建立维数间关系。
Gaussian upper bounds, volume doubling and Sobolev inequalities on graphs
本文研究图上Sobolev不等式与高斯热核上界及体积倍增条件的等价性，通过引入局部正则性条件和可变修正函数，建立了Sobolev维数与倍增维数及顶点度增长之间的联系。
Correlated Gaussian Moments Determine Polynomials up to Orthogonal Equivalence
该研究证明关联高斯副本的混合矩可分离多项式的正交轨道，构造了有限高斯重构方案，并将其推广至不规则周期系统，建立了轨道证书与周期的对应关系。
Co-spectral radius for countable equivalence relations
本文定义可数等价关系包含的共谱半径，推广2-3方法证明有界度单模随机有根图的游走增长存在，并研究其在超有限关系与渗流临界指数中的应用。
An equivalence between gauge-twisted and topologically conditioned scalar Gaussian free fields
本文证明度量图上规范扭曲的标量高斯自由场等价于对通常场进行拓扑条件作用，并给出拓扑事件概率的行列式比表达式，进而提出强度加倍猜想。
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。
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.
We study spectral problems for integro-differential equations arising in the theory of Gaussian processes similar to the fractional Brownian motion. We generalize the method of Chigansky--Kleptsyna and obtain the two-term eigenvalue asymptotics for such equations. Application to the small ball probabilities in $L_2$-norm is given.
Degenerate self-similar measures, spectral asymptotics and small deviations of Gaussian processes
本文证明了一类高斯过程的$L_2$ -范数关于退化自相似测度的对数小球渐近性，这类高斯过程包括布朗运动，Ornstein-Uhlenbeck过程及其积分过程.
The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles
导出了随机Jacobi矩阵的平均谱测度的一个显式公式。该矩阵可以被认为是高斯β系综（G $β$ E）矩阵的极限，因为矩阵大小$N$在$N β$是常数的约束下趋于无穷大。
We study the small ball asymptotics problem in $L_2$ for two generalizations of the fractional Brownian motion with variable Hurst parameter. To this end, we perform careful analysis of the singular values asymptotics for associated integral operators.
Law equivalence for Ornstein--Uhlenbeck dynamics driven by Lévy noise
该研究针对Lévy噪声驱动的Ornstein-Uhlenbeck动力学，扩展律比较结果至C₀半群生成元，证明路径律绝对连续性与等价性，给出纯跳跃噪声及特定半群的充分条件，明确关键要求的作用。
For random systems subject to a constraint, the microcanonical ensemble requires the constraint to be met by every realisation ("hard constraint"), while the canonical ensemble requires the constraint to be met only on average ("soft constraint"). It is known that for random graphs subject to topological constraints breaking of ensemble equivalence may occur when the size of the graph tends to infinity, signalled by a non-vanishing specific relative entropy of the two ensembles. We investigate to what extent breaking of ensemble equivalence is manifested through the largest eigenvalue of the a…
We prove that the spectral structure on the $N$-dimensional standard sphere of radius $(N-1)^{1/2}$ compatible with a projection onto the first $n$-coordinates converges to the spectral structure on the $n$-dimensional Gaussian space with variance $1$ as $N\to \infty$. We also show the analogue for the first Dirichlet eigenvalue and its eigenfunction on a ball in the sphere and on a half-space in the Gaussian space.
Asymptotic equivalence for inhomogeneous jump diffusion processes and white noise
我们证明了由离散（高频）或连续观察时间非均匀跳跃扩散过程的路径和高斯白噪声实验产生的实验之间的全局渐近等价性。这里，所考虑的参数是漂移函数，并且我们假设观测时间$T$趋向于$\infty$。近似的Le Cam $Δ$距离的意义下，光滑条件下的未知漂移函数。这些渐近等价性是通过构建显式马尔可夫核来建立的，这些显式马尔可夫核可以用来从另一个实验中重现一个实验。
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.
Transfer Operators and SRB Measures for Axiom A Diffeomorphisms: Spectral Gap, Structural Stability, and the Gibbs Equivalence Theorem
本文研究Axiom A微分同胚的转移算子和SRB测度，通过马尔可夫分区编码将符号谱结果推广到光滑动力学，提出四个主要定理，包括结构稳定性、转移算子的拟紧致性、SRB测度的构造以及佩辛熵公式，最终通过四个主要定理和导入结果证明吉布斯等价定理。
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