Shahar Mendelson, Joel Zinn
We show that a modified Empirical process converges to the limiting Gaussian process whenever the limit is continuous. The modification depends on the properties of the limit via Talagrand's characterization of the continuity of Gaussian processes.
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Drees and Rootzén [2010] have proven central limit theorems (CLT) for empirical processes of extreme values cluster functionals built from $β$-mixing processes. The problem with this family of $β$-mixing processes is that it is quite restrictive, as has been shown by Andrews [1984]. We expand this result to a more general dependent processes family, known as weakly dependent processes in the sense of Doukhan and Louhichi [1999], but in finite-dimensional convergence (fidis). We show an example where the application of the CLT-fidis is sufficient in several cases, including a small simulation o…
For a uniform process $\{ X_t: t\in E\}$ (by which $X_t $ is uniformly distributed on $(0,1)$ for $t\in E$) and a function $w(x)>0$ on $(0,1)$, we give a sufficient condition for the weak convergence of the empirical process based on $\{ w(x)(\mathbb{1}_{X_t\leq x} -x): t\in E, x\in [0,1]\}$ in $\ell^\infty(E\times [0,1])$. When specializing to $w(x)\equiv 1$ and assuming strict monotonicity on the marginal distribution functions of the input process, we recover a result of Kuelbs, Kurtz, and Zinn (2013). In the last section, we give an example of the main theorem.
A central limit theorem (CLT) for the smoothed empirical spectral distribution of sample covariance matrices is established. Moreover, the CLTs for the smoothed quantiles of Marcenko and Pastur's law have been also developed.
We establish empirical quantile process CLTs based on $n$ independent copies of a stochastic process $\{X_t: t \in E\}$ that are uniform in $t \in E$ and quantile levels $α\in I$, where $I$ is a closed sub-interval of $(0,1)$. Typically $E=[0,T]$, or a finite product of such intervals. Also included are CLT's for the empirical process based on $\{I_{X_t \le y} - \rm {Pr}(X_t \le y): t \in E, y \in R \}$ that are uniform in $t \in E, y \in R$. The process $\{X_t: t \in E\}$ may be chosen from a broad collection of Gaussian processes, compound Poisson processes, stationary independent increment …
In a paper of Jason Swanson, a CLT for the sample median of independent Brownian motions with value 0 at 0 was proved. Here we extend this result in two ways. We prove such a result for a collection of self-similar processes which include the fractional Brownian motions, all stationary, independent increment symmetric stable processes tied down at 0 as well as iterated and integrated Brownian motions. Second, our results hold uniformly over all quantiles in a compact sub-interval of (0,1). We also examine sample function properties connected with these CLTs.
We consider the spectral properties of a class of regularized estimators of (large) empirical covariance matrices corresponding to stationary (but not necessarily Gaussian) sequences, obtained by banding. We prove a law of large numbers (similar to that proved in the Gaussian case by Bickel and Levina), which implies that the spectrum of a banded empirical covariance matrix is an efficient estimator. Our main result is a central limit theorem in the same regime, which to our knowledge is new, even in the Gaussian setup.
For stochastic processes $\{X_t:t\in E\}$, we establish sufficient conditions for the empirical process based on $\{I_{X_t\le y}-\operatorname{Pr}(X_t\le y):t\in E,y\in\mathbb{R}\}$ to satisfy the CLT uniformly in $t\in E,y\in\mathbb{R}$. Corollaries of our main result include examples of classical processes where the CLT holds, and we also show that it fails for Brownian motion tied down at zero and $E=[0,1]$.
We prove central limit theorems (CLT) for empirical processes of extreme values cluster functionals as in Drees and Rootzén (2010). We use coupling properties enlightened for Dedecker \& Prieur's $τ-$dependence coefficients in order to improve the conditions of dependence and continue to obtain these CLT. The assumptions are precisely set for particular processes and cluster functionals of interest. The number of excesses provides a complete example of a cluster functional for a simple non-mixing model (AR(1)-process) for which ours results are definitely needed. We also give the expression ex…
Mean field games (MFGs) describe the limit, as $n$ tends to infinity, of stochastic differential games with $n$ players interacting with one another through their common empirical distribution. Under suitable smoothness assumptions that guarantee uniqueness of the MFG equilibrium, a form of law of large of numbers (LLN), also known as propagation of chaos, has been established to show that the MFG equilibrium arises as the limit of the sequence of empirical measures of the $n$-player game Nash equilibria, including the case when player dynamics are driven by both idiosyncratic and common sourc…
Customers arrive at rate N times alpha on a network of N single server infinite buffer queues, choose L queues uniformly, join the shortest one, and are served there in turn at rate beta. We let N go to infinity.We prove a functional central limit theorem (CLT) for the tails of the empirical measures of the queue occupations,in a Hilbert space with the weak topology, with limit given by an Ornstein-Uhlenbeck process. The a priori assumption is that the initial data converge.This completes a recent functional CLT in equilibrium result for which convergence for the initial data was not known in …
We consider "randomized" statistics constructed by using a finite number of observations a random field at randomly chosen points. We generalize the invariance principle (the functional CLT), the Glivenko--Cantelli theorem, the theorem about convergence to the Brownian bridge and the Kolmogorov theorem about the limit distribution of the empirical distribution function, as well as an improved version of the CLT in A. Tempelman, Randomized multivariate central limit theorems for ergodic homogeneous random fields, Stochastic Processes and their Applications. 143 (2022), 89-105. The randomized ap…
We show central limit theorems (CLT) for the Stieltjes transforms or more general analytic functions of symmetric matrices with independent heavy tailed entries, including entries in the domain of attraction of $α$-stable laws and entries with moments exploding with the dimension, as in the adjacency matrices of Erdös-Rényi graphs. For the second model, we also prove a central limit theorem of the moments of its empirical eigenvalues distribution. The limit laws are Gaussian, but unlike to the case of standard Wigner matrices, the normalization is the one of the classical CLT for independent r…
Statistical Inference for Adversarial Training: Central Limit Theorems via Optimal Transport
本文通过最优传输理论严格推导对抗训练模型的中心极限定理，证明最优势唯一性，并给出鞍点稳定性与泛化误差的样本复杂度。
We consider the macroscopic large N limit of the Circular beta-Ensemble at high temperature, and its weighted version as well, in the regime where the inverse temperature scales as beta/N for some parameter beta>0. More precisely, in the large N limit, the equilibrium measure of this particle system is described as the unique minimizer of a functional which interpolates between the relative entropy (beta=0) and the weighted logarithmic energy (beta=\infty). More precisely, we provide subGaussian concentration estimates in the W1 metric for the deviations of the empirical measure to this equili…
We derive limit theorems for the empirical distribution function of "devolatilized" increments of an Itô semimartingale observed at high frequencies. These "devolatilized" increments are formed by suitably rescaling and truncating the raw increments to remove the effects of stochastic volatility and "large" jumps. We derive the limit of the empirical c.d.f. of the adjusted increments for any Itô semimartingale whose dominant component at high frequencies has activity index of $1<β\le2$, where $β=2$ corresponds to diffusion. We further derive an associated CLT in the jump-diffusion case. We use…
Existence and non-existence of the CLT for a family of SDEs driven by stable process
本文研究了一类由α-稳定过程驱动且无全局Lipschitz漂移的SDE，在特定条件下证明了其经验测度正态中心极限定理的存在性与非存在性。
We consider the problem of optimal transportation with quadratic cost between a empirical measure and a general target probability on R d , with d $\ge$ 1. We provide new results on the uniqueness and stability of the associated optimal transportation potentials , namely, the minimizers in the dual formulation of the optimal transportation problem. As a consequence, we show that a CLT holds for the empirical transportation cost under mild moment and smoothness requirements. The limiting distributions are Gaussian and admit a simple description in terms of the optimal transportation potentials.
We establish a multivariate empirical process central limit theorem for stationary $\R^d$-valued stochastic processes $(X_i)_{i\geq 1}$ under very weak conditions concerning the dependence structure of the process. As an application we can prove the empirical process CLT for ergodic torus automorphisms. Our results also apply to Markov chains and dynamical systems having a spectral gap on some Banach space of functions. Our proof uses a multivariate extension of the techniques introduced by Dehling, Durieu and Volný \cite{DehDurVol09} in the univariate case. As an important technical ingredien…
It has recently been shown that there are substantial differences in the regularity behavior of the empirical process based on scalar diffusions as compared to the classical empirical process, due to the existence of diffusion local time. Besides establishing strong parallels to classical theory such as Ossiander's bracketing CLT and the general Giné-Zinn CLT for uniformly bounded families of functions, we find increased regularity also for multivariate ergodic diffusions, assuming that the invariant measure is finite with Lebesgue density $π$. The effect is diminishing for growing dimension b…
It is well-known that, on a purely algebraic level, a simplified algebraic version of the Central Limit Theorem (CLT) can be proved in the framework of a noncommutative probability space, under the hypotheses that the sequence of non-commutative random variables we consider is exchangeable and obeys a certain vanishing condition of some of its joint moments. In this approach (which covers versions for both the classical CLT and the CLT of free probability), the determination of the resulting limit law has to be addressed on a case-by-case basis. In this paper we discuss an instance of the abov…
Wasserstein-p Bounds in the Central Limit Theorem Under Local Dependence
本文将独立随机变量在Wasserstein-p距离下的中心极限定理最优误差界推广至局部依赖随机变量（如m-依赖随机场和U-统计量），通过改进Stein依赖邻域法推导最优收敛速率并给出尾界应用。
In this paper, we study the spatial averages of the solution to the parabolic Anderson model driven by a space-time Gaussian homogeneous noise that is colored in time and space. We establish quantitative central limit theorems (CLT) of this spatial statistics under some mild assumptions, by using the Malliavin-Stein approach. The highlight of this paper is the obtention of rate of convergence in the colored-in-time setting, where one can not use Itô's calculus due to the lack of martingale structure. In particular, modulo highly technical computations, we apply a modified version of second-ord…
Limit theorems for moment processes of beta Dyson's Brownian motions and beta Laguerre processes
本文研究了参数beta与系统尺寸倒数成正比时，beta Dyson Brown运动和beta Laguerre过程矩过程的动态极限定理，建立了相应的LLN和CLT。
Propagation of Chaos and Gaussian Fluctuations for Mean-field Coupled Maps
该研究针对平均场耦合映射对应的离散时间相互作用粒子系统，建立了相关估计与CLT型结果，证明混沌传播及经验测度波动依分布收敛于高斯过程，兼具定性与定量结论。
Efficient Concentration with Gaussian Approximation
针对有界随机变量，提出具有渐近最优宽度、有限样本有效性和次高斯衰减的可计算集中不等式，通过零偏耦合和Stein交换对方法紧界非均匀Kolmogorov和Wasserstein距离，构建高效置信区间和经验Berry-Esseen界。
We study the modified log-Sobolev inequality for a class of pure jump Markov processes that describe the interactions between brain neurons. In particular, we focus on a finite and compact process with degenerate jumps inspired by the model introduced by Galves and Löcherbach. As a result, we obtain concentration properties for empirical approximations of the process.
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