A simple invariance theorem
Sourav Chatterjee
本文对Lindeberg关于中心极限定理的论证作了一个单扩张，得到了一个一般的不变性结果。我们应用该技术来证明随机矩阵理论，自旋玻璃，和随机场的最大值的结果。
We present a simple extension of Lindeberg's argument for the Central Limit Theorem to get a general invariance result. We apply the technique to prove results from random matrix theory, spin glasses, and maxima of random fields.
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We present a Cameron--Martin type quasi-invariance theorem for subordinate Brownian motion. As applications, we establish an integration by parts formula and construct a gradient operator on the path space of subordinate Brownian motion, and we obtain some canonical Dirichlet forms. These findings extend the corresponding classical results for Brownian motion.
In 1983, N. Herrndorf proved that for a $ϕ$-mixing sequence satisfying the central limit theorem and $\liminf_{n\to\infty}\frac{σ^2_n}n>0$, the weak invariance principle takes place. The question whether for strictly stationary sequences with finite second moments and a weaker type ($α$, $β$, $ρ$) of mixing the central limit theorem implies the weak invariance principle remained open. We construct a strictly stationary $β$-mixing sequence with finite moments of any order and linear variance for which the central limit theorem takes place but not the weak invariance principle.
We provide a permutation-invariant version of the Komlós' theorem for non-negative random variables. The proof is quite elementary in the sense that it did not use the Axiom of Choice, and was based on a recent result in [3].
Density of eigenvalues and its perturbation invariance in unitary ensembles of random matrices
一般由正交多项式的正则变条件下的递推系数来研究随机矩阵酉系综的特征值密度。首先我们直接计算密度的矩。然后，通过研究矩的某些变形，得到了密度满足的一类一阶微分方程（见定理1.2），并通过求解这些微分方程给出了密度。进一步，我们证明了密度在权函数的多项式扰动后是不变的（见定理1.5）。
A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation
我们表明，经典的de Finetti定理有一个典型的非交换对应，如果我们加强'交换'（即，随机变量的联合分布在置换群作用下的不变性）到在量子置换群作用下的不变性。更确切地说，对于一个无穷序列的非交换随机变量，我们证明了他们的联合分布量子置换下的不变性等价于这样一个事实，即随机变量是同分布的和自由的关于条件期望到他们的尾代数。
Nonlinear effects within invariance principles
本文从统一视角给出Green-Kubo型精确公式，刻画快慢系统多变量分析中布朗粗糙路径极限，并讨论其在不同情境下的应用。
We prove the Lukacs-Olkin-Rubin theorem without invariance of the distribution of the "quotient", which was the key assumption in the original proof of [Olkin--Rubin, Ann. Math. Stat. 33 (1962), 1272--1280]. Instead we assume existence of strictly positive continuous densities of respective random variables. We consider the (cone variate) "quotient" for any division algorithm satisfying some natural conditions. For that purpose, the new proof of the Olkin--Baker functional equation on symmetric cones is given.
The investigation asymptotic limits on associated data mainly focused on limit theorems of summands of associated data and on the related invariance principles. In a series of papers, we are going to set the general frame of the theory by considering an arbitrary infinitely decomposable (divisible) limit law for summands and study the associated functional laws converging to Lévy processes. The asymptotic frame of Newman (1980) is still used as a main tool. Detailed results are given when $G$ is a Gaussian law (as confirmation of known results) and when $G$ is a Poisson law. In the later case,…
In earlier work we showed that the particle displacement for the multidimensional periodic Lorentz gas, in the limit of low scatterer density (Boltzmann-Grad limit), satisfies a central limit theorem with superdiffusive scaling. The present paper extends this result to a functional central limit theorem, i.e., the weak convergence of the particle trajectory to Brownian motion.
Quenched invariance principle for simple random walk on percolation clusters
本文考虑了含$d\ge2$的$\Z^d$模型中超临界键渗流（唯一）无限集团上的简单随机游动。我们证明，几乎每一个渗流配置，步行的路径分布弱收敛到非退化，各向同性布朗运动。我们的分析是基于考虑的调和变形的无限集群上的随机游动成为一个平方可积鞅。变形的大小，表示为所谓的校正，估计通过遍历参数。
Shift and stretch invariance lead to the exponential-Boltzmann probability distribution. Rotational invariance generates the Gaussian distribution. Particular scaling relations transform the canonical exponential and Gaussian patterns into the variety of commonly observed patterns. The scaling relations themselves arise from the fundamental invariances of shift, stretch, and rotation, plus a few additional invariances. Prior work described the three fundamental invariances as a consequence of the equilibrium canonical ensemble of statistical mechanics or the Jaynesian maximization of informati…
The simple Lévy Poisson process and scaled forms are explicitly constructed from partial sums of independent and identically distributed random variables and from sums of non-stationary independent random variables. For the latter, the weak limits are scaled Poisson processes. The method proposed here prepares generalizations to dependent data, to associated data in the first place.
We address the problem of testing for the invariance of a probability measure under the action of a group of linear transformations. We propose a procedure based on consideration of one-dimensional projections, justified using a variant of the Cramér-Wold theorem. Our test procedure is powerful, computationally efficient, and dimension-independent, extending even to the case of infinite-dimensional spaces (multivariate functional data). It includes, as special cases, tests for exchangeability and sign-invariant exchangeability. We compare our procedure with some previous proposals in these cas…
A Nonconventional Invariance Principle for Random Fields
证明了随机场的一个非常规的不变原理（泛函中心极限定理）。
By a classical principle of probability theory, sufficiently thin subsequences of general sequences of random variables behave like i.i.d.\ sequences. This observation not only explains the remarkable properties of lacunary trigonometric series, but also provides a powerful tool in many areas of analysis, such the theory of orthogonal series and Banach space theory. In contrast to i.i.d.\ sequences, however, the probabilistic structure of lacunary sequences is not permutation-invariant and the analytic properties of such sequences can change after rearrangement. In a previous paper we showed t…
General Central limit theorem deals with weak limits (in type) of sums of row-elements of array random variables. In some situations as in the invariance principle problem, the sums may include only parts of the row-elements. For strictly stationary arrays (stationary for each row), there is no change to the asymptotic results. But for non-stationary data, especially for dependent data, asymptotic laws of partial sums moving in rows may require extra-conditions to exist. This paper deals with central limit theorems with Gaussian limits for non-stationary data. Our main focus is on dependent da…
Quenched Invariance Principles via Martingale Approximation
本文综述了平稳过程和遍历过程的几乎必然中心极限定理及其函数形式。对于平稳和遍历马尔可夫链的可加泛函，这些定理在中心极限定理及其从一点开始的泛函形式下是已知的。所有这些结果的共同点是它们都是通过几乎必然意义下的鞅逼近得到的。我们指出了几个应用这些结果类的混合序列，移位过程，可逆马尔可夫链，Metropolis Hastings算法。
The invariance principle for inhomogeneous Diophantine approximations
本文利用累积量方法，通过仿射格点空间上高阶相关性的有效混合，证明了非齐次丢番图逼近的中心极限定理与不变性原理。
We prove an invariance principle for functions on a slice of the Boolean cube, which is the set of all vectors {0,1}^n with Hamming weight k. Our invariance principle shows that a low-degree, low-influence function has similar distributions on the slice, on the entire Boolean cube, and on Gaussian space. Our proof relies on a combination of ideas from analysis and probability, algebra and combinatorics. Our result imply a version of majority is stablest for functions on the slice, a version of Bourgain's tail bound, and a version of the Kindler-Safra theorem. As a corollary of the Kindler-Safr…
In this paper, we investigate the functional central limit theorem for stochastic processes associated to partial sums of additive functionals of reversible Markov chains with general spate space, under the normalization standard deviation of partial sums. For this case, we show that the functional central limit theorem is equivalent to the fact that the variance of partial sums is regularly varying with exponent 1 and the partial sums satisfy the CLT. It is also equivalent to the conditional CLT.
Almost sure functional central limit theorem for non-nestling random walk in random environment
考虑乘积随机环境中的非雏型随机游动。我们假设行走的一步是指数矩，在环境中是均匀的。我们证明了一个不变原理（功能中心极限定理）几乎在每一个环境下的中心和扩散缩放行走。不变原理背后的要点是，游走的猝灭平均值表现为亚扩散。
A note on the invariance principle of the product of sums of random variables
文献中研究了各种随机变量和乘积的中心极限定理。本文旨在说明这类结果是不变原理的推论。
A New Condition for the Invariance Principle for Stationary Random Fields
本文建立了具有射影型条件的平稳随机场的中心极限定理和不变原理。我们的结果是通过一个m依赖的近似方法。作为应用，我们建立了线性随机场的正交鞅和泛函的不变原理。
Consider a Bernoulli random field satisfying the Hannan's condition. Recently, invariance principles for partial sums of random fields over rectangular index sets are established. In this note we complement previous results by investigating limit theorems for weighted Bernoulli random fields, including central limit theorems for partial sums over arbitrary index sets and invariance principles for Gaussian random fields. Most results improve earlier ones on Bernoulli random fields under Wu's condition, which is stronger than Hannan's condition.
Almost sure functional central limit theorem for ballistic random walk in random environment
考虑乘积随机环境中的多维随机游动，其步数有界，在某个空间方向上具有瞬时性，再生时刻具有足够高的矩。我们证明了一个不变原理，或功能中心极限定理，几乎在每一个环境下的扩散缩放中心行走。不变原理背后的主要观点是，行走的淬火平均表现为次扩散。
We introduce a family of quantum semigroups and their natural coactions on noncommutative polynomials. We present three invariance conditions, associated with these coactions, for the joint distribution of sequences of selfadjoint noncommutative random variables. For one of the invariance conditions, we prove that the joint distribution of an infinite sequence of noncommutative random variables satisfies it is equivalent to the fact that the sequence of the random variables are identically distributed and boolean independent with respect to the conditional expectation onto its tail algebra. Th…
An Almost Sure Invariance Principle for Additive Functionals of Markov Chains
证明了马尔可夫链的向量值可加泛函关于遍历平衡分布对几乎所有起点的不变原理（泛函中心极限定理）。这个假设是一个约束在预解式上的矩。
In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general R d-valued cocycles that appear in the study of the left random walk on linear groups. Our method of proof lies on a suitable martingale approximation and on a careful estimation of some coupling coefficients linked with the underlying Markov structure. Concerning the martingale part, the available results in the literature are not accurate enough to give almost optimal rates whether in the central limit theorem for the Wasserstein distance, or in the strong approximation.…
Quenched invariance principle for simple random walk on discrete point processes
本文考虑离散点过程生成的随机图上的简单随机游动。该随机图具有立方晶格的随机子集作为顶点，并且平行于每个坐标轴的线上的任何连续顶点之间的线作为边。在离散点过程是平稳的、依赖于时间的假设下，我们证明了猝灭不变原理成立，即对几乎每一个点过程的构形，其游动的路径分布都弱收敛于布朗运动的路径分布.
A strong invariance principle for nonconventional sums
本文研究了$\sum_{n=1}^{[Nt]}F\big(X(n),X(2n),...,X(kn), X(q_{k+1}(n)),X(q_{k+2}(n)),..., X(q_\ell(n))\big)$型和的强不变原理（也称为强逼近定理
Almost Sure Invariance Principle for Continuous-Space Random Walk in Dynamic Random Environment
我们考虑了在多项式混合随机环境中$\R^d$上的随机游动，该环境在每个时间步都被刷新。利用鞅方法给出了几乎处处泛函中心极限定理成立的一个充要条件。
We study the asymptotic behaviour of random walks in i.i.d. non-elliptic random environments on $\mathbb{Z}^d$. Standard conditions (and proofs) for ballisticity and the central limit theorem require ellipticity. We use oriented percolation and martingale arguments to give non-trivial local conditions for ballisticity and an annealed invariance principle in the non-elliptic setting.
We prove a quenched central limit theorem for balanced random walks in time dependent ergodic random environments which is not necessarily nearest-neigbhor. We assume that the environment satisfies appropriate ergodicity and ellipticity conditions. The proof is based on the use of a maximum principle for parabolic difference operators.
A quenched weak invariance principle
本文研究了一类满足投影准则的随机变量的几乎几乎必然条件中心极限定理的泛函形式。给出了强混合过程和不可约马尔可夫链的应用。证明是基于正常的双指标鞅样序列，理论本身有兴趣的近似。
We prove quenched invariance principle for simple random walk on the unique infinite percolation cluster for a general class of percolation models on Z^d, d>=2, with long-range correlations introduced in arXiv:1212.2885, solving one of the open problems from there. This gives new results for random interlacements in dimension d>=3 at every level, as well as for the vacant set of random interlacements and the level sets of the Gaussian free field in the regime of the so-called local uniqueness (which is believed to coincide with the whole supercritical regime). An essential ingredient of our pr…
A probabilistic proof of the Open Mapping Theorem for analytic functions
利用布朗运动的共形不变性给出了解析函数的开映射定理的一个简短证明。
Strassen's invariance principle for random walk in random environment
本文考虑了$\mathbb{Z}^{d}\,(d\geq1)$上随机环境中的随机游动，利用鞅论和Derriennic和Lin \cite{DL}的分数余边界理论，在一定条件下证明了该模型的Strassen强不变原理.所得结果部分地填补了大数定律与中心极限定理之间的空白。
Invariance Principle for the Random Conductance Model with dynamic bounded Conductances
研究了动态随机电导环境中的连续时间随机游动X。我们假设电导是平稳遍历的，一致有界的，有界远离零和多项式混合的空间和时间。证明了X的猝灭不变原理，得到了格林函数的界和局部极限定理.我们还讨论了随机接口模型的连接。
On martingale approximations and the quenched weak invariance principle
本文利用投影准则给出了在${\mathcal{H}}$（一个实的可分希尔伯特空间）中平稳过程的部分和在${\mathbb{L}}^p({\mathcal{H}})$，$p>1$中可由平稳差鞅逼近的充分条件，并估计了在${\mathbb{L}}^p({\mathcal{H}})$中逼近的误差.结果被用来进一步研究部分和的行为。特别地，我们得到了关于Marcinkiewicz-Zygmund定理，中偏差原理和中心极限定理中的速率的新的投影条件.这些条件非常适合于各种各样的例子，包括线性过程或各种弱相关或混合过程。此外，我们的方法适合通过鞅近似来研究淬火中心极限定理及其不变原理，并允许我们证明它们在所谓的Maxwell-Woodroofe条件下是最优的。
Invariance principle and local limit theorem for a class of random conductance models with long-range jumps
本文研究了允许长程跳转的随机电导模型上的连续时间随机游走，在最优矩条件下证明了淬灭不变性原理与局部极限定理，并得到了热方程解的正则性估计及谱维度结论。
Cyclic structure and dynamics are of great interest in both the fields of stochastic processes and nonequilibrium statistical physics. In this paper, we find a new symmetry of the Brownian motion named as the quasi-time-reversal invariance. It turns out that such an invariance of the Brownian motion is the key to prove the cycle symmetry for diffusion processes on the circle, which says that the distributions of the forming times of the forward and backward cycles, given that the corresponding cycle is formed earlier than the other, are exactly the same. With the aid of the cycle symmetry, we …
An invariance principle for fractional Brownian sheets
建立了具有相依新息的平稳线性随机场部分和的中心极限定理和各向异性分数布朗单的不变原理。我们的结果是Dedecker等人（2011）的分数布朗运动的不变原理在高维的推广。他们的论点的一个关键组成部分，鞅近似，被替换为m-近似参数。我们的方法的一个重要工具是最近由El Machkouri et al.（2011）建立的平稳随机场的矩不等式。
A functional limit theorem for lattice oscillating random walk
本文研究了 $\mathbb{Z}$ 上 Kemperman 振荡随机游走的不变性原理，通过利用考虑正负半轴间振荡的 Markov 子过程构造非周期更新算子序列，并结合 S. Gouëzel 的定理，推广了经典与反射随机游走的结果。
A continuous analogue of the invariance principle and its almost sure version
我们处理的随机过程从一个齐次随机独立增量过程通过更换的时间尺度和乘以一个赋范常数。在Skorokhod空间中，我们证明了这些过程在一致收敛的一致收敛拓扑下依分布收敛于Wiener过程。得到了该定理的一个积分型几乎必然形式。
Recent advances in invariance principles for stationary sequences
本文综述了平稳序列的中心极限定理及其弱不变原理的一些最新结果。我们还描述了几个极大不等式，它们是获得不变原理的主要工具，而且它们本身也很有趣。所考虑的相依随机变量类有类鞅序列、混合序列、线性过程、遍历马尔可夫链的可加泛函。
The Newcomb-Benford law, also known as the first-digit law, gives the probability distribution associated with the first digit of a dataset, so that, for example, the first significant digit has a probability of $30.1$ % of being $1$ and $4.58$ % of being $9$. This law can be extended to the second and next significant digits. This article presents an introduction to the discovery of the law, its derivation from the scale invariance property, as well as some applications and examples, are presented. Additionally, a simple model of a Markov process inspired by scale invariance is proposed. With…
We obtain an elementary invariance principle for multi-dimensional Brownian sheet where the underlying random fields are not necessarily independent or stationary. Possible applications include unit-root tests for spatial as well as panel data models.
Based on deleting-item central limit theory, the classical Donsker's theorem of partial-sum process of independent and identically distributed (i.i.d.) random variables is extended to incomplete partial-sum process. The incomplete partial-sum process Donsker's invariance principles are constructed and derived for general partial-sum process of i.i.d random variables and empirical process respectively, they are not only the extension of functional central limit theory, but also the extension of deleting-item central limit theory. Our work enriches the random elements structure of weak convergen…
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