On the concentration of measure phenomenon for stable and related random vectors

随机动力系统的集中现象：一种算子理论方法

Concentration Phenomenon for Random Dynamical Systems: An Operator Theoretic Approach

本文利用算子理论方法研究马尔可夫链无界观测量/奖励函数的集中现象，通过转移算子与乘法算子的复合及超有界性条件，在无需系统精确信息和可逆性假设下建立了尖锐的非渐近集中界，对强化学习与控制领域具有重要应用价值。

Concentration of measure via approximated Brunn--Minkowski inequalities

We prove that an approximated version of the Brunn--Minkowski inequality with volume distortion coefficient implies a Gaussian concentration-of-measure phenomenon. Our main theorem is applicable to discrete spaces.

Higher Order Concentration of Measure

We study sharpened forms of the concentration of measure phenomenon typically centered at stochastic expansions of order $d-1$ for any $d \in \mathbb{N}$. The bounds are based on $d$-th order derivatives or difference operators. In particular, we consider deviations of functions of independent random variables and differentiable functions over probability measures satisfying a logarithmic Sobolev inequality, and functions on the unit sphere. Applications include concentration inequalities for $U$-statistics as well as for classes of symmetric functions via polynomial approximations on the sphe…

Central and $L^p$-concentration of 1-Lipschitz maps into $\mathbb{R}$-trees

In this paper, we study the Lévy-Milman concentration phenomenon of 1-Lipschitz maps from mm-spaces to $\mathbb{R}$-trees. Our main theorems assert that the concentration to $\mathbb{R}$-trees is equivalent to the concentration to the real line.

Talagrand Concentration Inequalities for Stochastic Partial Differential Equations

One way to define the concentration of measure phenomenon is via Talagrand inequalities, also called transportation-information inequalities. That is, a comparison of the Wasserstein distance from the given measure to any other absolutely continuous measure with finite relative entropy. Such transportation-information inequalities were recently established for some stochastic differential equations. Here, we develop a similar theory for some stochastic partial differential equations.

Concentration of measure without independence: a unified approach via the martingale method

The concentration of measure phenomenon may be summarized as follows: a function of many weakly dependent random variables that is not too sensitive to any of its individual arguments will tend to take values very close to its expectation. This phenomenon is most completely understood when the arguments are mutually independent random variables, and there exist several powerful complementary methods for proving concentration inequalities, such as the martingale method, the entropy method, and the method of transportation inequalities. The setting of dependent arguments is much less well unders…

Matrix Poincaré inequalities and concentration

We show that any probability measure satisfying a Matrix Poincaré inequality with respect to some reversible Markov generator satisfies an exponential matrix concentration inequality depending on the associated matrix carré du champ operator. This extends to the matrix setting a classical phenomenon in the scalar case. Moreover, the proof gives rise to new matrix trace inequalities which could be of independent interest. We then apply this general fact by establishing matrix Poincaré inequalities to derive matrix concentration inequalities for Gaussian measures, product measures and for Strong…

Second order concentration on the sphere

Sharpened forms of the concentration of measure phenomenon for classes of functions on the sphere are developed in terms of Hessians of these functions.

On the rate of concentration of maxima in Gaussian arrays

Recently in Gao and Stoev (2018) it was established that the concentration of maxima phenomenon is the key to solving the exact sparse support recovery problem in high dimensions. This phenomenon, known also as relative stability, has been little studied in the context of dependence. Here, we obtain bounds on the rate of concentration of maxima in Gaussian triangular arrays. These results are used to establish sufficient conditions for the uniform relative stability of functions of Gaussian arrays, leading to new models that exhibit phase transitions in the exact support recovery problem. Fina…

Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

We consider the Markov chain $\{X_n^x\}_{n=0}^\infty$ on $\R^d$ defined by the stochastic recursion $X_{n}^{x}=\p_{θ_{n}}(X_{n-1}^{x})$, starting at $x\in\R^d$, where $θ_{1}, θ_{2},...$ are i.i.d. random variables taking their values in a metric space $(Θ, \mathfrak{r}),$ and $\p_{θ_{n}}:\R^d\mapsto\R^d$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure $ν$. Under appropriate assumptions on $\p_{θ_n}$, we will show that the measure $ν$ has a heavy tail with the exponent $α>0$ i.e. $ν(\{x\in\R^d: |x|>t\})\asymp t^{-α}$. Using this result we show that properly norm…

浓度测度现象的一个推广及其在交集问题中的应用

A Generalisation of the Concentration-of-Measure Phenomenon with Applications to Intersection Problems

本文在离散立方体上推广了浓度测度现象，定义了“(γ,C)-良好散布”子集，并证明对于任何大子集A和良好散布集S，A+S的测度接近全空间。利用该结果，对C4-交集问题和4-AP-交集问题给出了非平凡上界，并对H-交集问题和k-AP-交集问题给出了趋于0的上界。

小Lévy噪声下非线性稳定朗之万系统在Wasserstein距离中的截止现象

The cutoff phenomenon in Wasserstein distance for nonlinear stable Langevin systems with small Lévy noise

该研究将小噪声下非线性稳定系统的截止现象结论推广至Wasserstein距离框架，覆盖一般Lévy噪声情形，通过精细控制Hartman-Grobman定理误差与适配线性理论完成证明，并给出相关物理系统示例。

关于随机排列的反集中现象

The anti-concentration phenomenon with respect to random permutations

本文研究了在对称群上随机排列的反集中现象，分析了离散和连续情况下的集中概率，并应用结果到随机多项式和随机矩阵中。

Long-term concentration of measure and cut-off

We present new concentration of measure inequalities for Markov chains, generalising results for chains that are contracting in Wasserstein distance. These are particularly suited to establishing the cut-off phenomenon for suitable chains. We apply our discrete-time inequality to the well-studied Bernoulli-Laplace model of diffusion, and give a probabilistic proof of cut-off, recovering and improving the bounds of Diaconis and Shahshahani. We also extend the notion of cut-off to chains with an infinite state space, and illustrate this in a second example, of a two-host model of disease in cont…

含小幅度层状稳定噪声的非线性朗之万系统在全变差意义下的截断现象

The cutoff phenomenon in total variation for nonlinear Langevin systems with small layered stable noise

本文研究小幅度层状稳定噪声扰动下的单稳态非线性朗之万系统，在全变差距离下证明其指数遍历性与截断现象，推广了相关结果并给出可验证的充分条件，证明技巧异于高斯情形。

Large deviation principle of occupation measure for stochastic real Ginzburg-Landau equation driven by $α$-stable noises

We establish a large deviation principle for the occupation measure of the stochastic real Ginzburg-Landau equation driven by $α$-stable noises. The proof is based on a hyper-exponential recurrence criterion. Our result indicates a phenomenon that strong dissipation beats heavy tailed noises to produce a large deviation, it seems to us that this phenomenon has not been reported in the known literatures.

高维次高斯向量的集中性

Concentration of a high dimensional sub-gaussian vector

本文研究高维次高斯向量经线性算子作用后平方范数的集中现象，利用最新 Laplace 逼近方法将高斯偏差界推广到非高斯情形，并给出独立同分布和情形的说明。

非线性随机Laplace方程：大偏差与测度集中

Nonlinear stochastic Laplace equation: Large Deviations and Measure Concentration

本文利用弱收敛方法和局部化时间增量估计，建立了无界域上由小乘性Brown噪声驱动的p-Laplace方程强解的大偏差原理，并基于Girsanov变换和L2-唯一性方法证明了二次传输成本信息不等式，导出测度集中现象。

The concentration of zero-noise limits of invariant measures for stochastic dynamical systems

In this paper, we study concentration phenomena of zero-noise limits of invariant measures for stochastic differential equations defined on $\mathbb{R}^d$ with locally Lipschitz continuous coefficients and more than one ergodic state. Under some dissipative conditions, by using Lyapunov-like functions and large deviations methods, we estimate the invariant measures in neighborhoods of stable sets, neighborhoods of unstable sets and their complement, respectively. Our result illustrates that invariant measures concentrate on the intersection of stable sets where a cost functional $W(K_i)$ is mi…

Concentration of Measure and Isoperimetric Inequalities in Product Spaces

The concentration of measure prenomenon roughly states that, if a set $A$ in a product $Ω^N$ of probability spaces has measure at least one half, ``most'' of the points of $Ω^N$ are ``close'' to $A$. We proceed to a systematic exploration of this phenomenon. The meaning of the word ``most'' is made rigorous by isoperimetric-type inequalities that bound the measure of the exceptional sets. The meaning of the work ``close'' is defined in three main ways, each of them giving rise to related, but different inequalities. The inequalities are all proved through a common scheme of proof. Remarkably, …

简单随机张量的尖锐集中性 II：非对称性

Sharp Concentration of Simple Random Tensors II: Asymmetry

本文研究了简单随机张量的尖锐集中不等式，发现非对称张量在有效秩跨越临界阈值时会出现额外对数因子。通过扩展通用链技术到高阶设置，建立了非对称张量的集中界。

熵-瓦瑟斯坦正则化，缺陷性局部集中与超越非负曲率的截断准则

Entropy-Wasserstein regularization, defective local concentration and a cutoff criterion beyond non-negative curvature

本文提出一种放松的Ollivier粗黎曼曲率变种，通过熵-运输正则化效应和缺陷性局部集中，推导出在负曲率设置中截断现象的出现准则。

存在浓度点时具有Markov切换的随机结构随机动态系统的稳定性

Stability of stochastic dynamic systems of a random structure with Markov switchings in the presence of concentration points

本文研究了存在浓度点时具有Markov切换的随机结构随机微分方程的稳定性，利用Lyapunov函数及对跳跃幅度和时间的约束证明了渐近稳定性，并通过实例验证。

On the tightness of Gaussian concentration for convex functions

The concentration of measure phenomenon in Gauss' space states that every $L$-Lipschitz map $f$ on $\mathbb R^n$ satisfies \[ γ_{n} \left(\{ x : | f(x) - M_{f} | \geqslant t \} \right) \leqslant 2 e^{ - \frac{t^2}{ 2L^2} }, \quad t>0, \] where $γ_{n} $ is the standard Gaussian measure on $\mathbb R^{n}$ and $M_{f}$ is a median of $f$. In this work, we provide necessary and sufficient conditions for when this inequality can be reversed, up to universal constants, in the case when $f$ is additionally assumed to be convex. In particular, we show that if the variance ${\rm Var}(f)$ (with respect t…

Random polynomials of high degree and Levy concentration of measure

We show that the L^p norms of random sequences {s_N} of L^2 normalized holomorphic sections of increasing powers of an ample line bundle on a compact Kahler manifold are almost surely bounded for 2<p< infinity, and are almost surely O((log N)^{1/2}) for p= infinity. This estimate also holds for almost-holomorphic sections of positive line bundles on symplectic manifolds (in the sense of math.SG/0212180) and we give almost sure bounds for the C^k norms. Our methods involve asymptotics of Bergman-Szego kernels and the concentration of measure phenomenon.

对称 α-稳定随机矩阵的受限拟凸等距性质

Restricted Quasiconvexity Isometry Property for Symmetric $α$-Stable Random Matrices

本文提出对称 α-稳定随机矩阵的受限拟凸等距性质（RQIP），通过分数矩集中不等式和稀疏 ℓ_α 球覆盖数界推导出行数下界，揭示多项式尾部样本复杂度，并说明 RIP 框架在实践中或需被零空间性质等方法替代。

Concentration of Measure Techniques and Applications

Concentration of measure is a phenomenon in which a random variable that depends in a smooth way on a large number of independent random variables is essentially constant. The random variable will "concentrate" around its median or expectation. In this work, we explore several theories and applications of concentration of measure. The results of the thesis are divided into three main parts. In the first part, we explore concentration of measure for several random operator compressions and for the length of the longest increasing subsequence of a random walk evolving under the asymmetric exclus…