Alexander Soshnikov
We study large Wigner random matrices in the case when the marginal distributions of matrix entries have heavy tails. We prove that the largest eigenvalues of such matrices have Poisson statistics.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove that an adequately rescaled sequence $\{F_n\}$ of self-adjoint operators, living inside a fixed free Wigner chaos of even order, converges in distribution to a centered free Poisson random variable with rate $λ>0$ if and only if $φ(F_n^4)-2φ(F_n^3)\rightarrow2λ^2-λ$ (where $φ$ is the relevant tracial state). This extends to a free setting some recent limit theorems by Nourdin and Peccati [Ann. Probab. 37 (2009) 1412-1426] and provides a noncentral counterpart to a result by Kemp et al. [Ann. Probab. 40 (2012) 1577-1635]. As a by-product of our findings, we show that Wigner chaoses of …
We prove a large deviation principle for the largest eigenvalue of Wigner matrices without Gaussian tails, namely such that the distribution tails $\mathbb{P}( |X_{1,1}|>t)$ and $\mathbb{P}(|X_{1,2}|>t)$ behave like $e^{-bt^α}$ and $e^{-at^α}$ respectively for some $a,b\in (0,+\infty)$ and $α\in (0,2)$. The large deviation principle is of speed $N^{α/2}$ and with a good rate function depending only on the tail distribution of the entries.
We find necessary and sufficient conditions for the free additive infinite divisibility of some free multiplicative convolutions with the Wigner, the arcsine, the free Poisson and other distributions, including explicit examples.
Based on recent findings by Bourguin and Peccati, we give a fourth moment type condition for an element of a free Poisson chaos of arbitrary order to converge to a free (centered) Poisson distribution. We also show that free Poisson chaos of order strictly greater than one do not contain any non-zero free Poisson random variables. We are also able to give a sufficient and necessary condition for an element of the first free Poisson chaos to have a free Poisson distribution. Finally, depending on the parity of the considered free Poisson chaos, we provide a general counterexample to the naive u…
Tensorial free convolution, semicircular, free Poisson and R-transform in high order
本文为紧支撑测度建立张量自由卷积，定义高阶半圆律与自由泊松律，证明Wishart型张量收敛，并引入R变换及首批例子。
In this paper we analyze the covariance kernel of the Gaussian process that arises as the limit of fluctuations of linear spectral statistics for Wigner matrices with a few moments. More precisely, the process we study here corresponds to Hermitian matrices with independent entries that have $α$ moments for $2<α< 4$. We obtain a closed form $α$-dependent expression for the covariance of the limiting process resulting from fluctuations of the Stieltjes transform by explicitly integrating the known double Laplace transform integral formula obtained in the literature. We then express the covarian…
We derive upper bounds on the tail conditional expectation of binomial and Poisson random variables. Those upper bounds are subsequently employed to the problem of obtaining non-asymptotic lower bounds on the probability that the aforementioned random variables are significantly larger than their expectation.
Let $X_N$ be an $N\ts N$ random symmetric matrix with independent equidistributed entries. If the law $P$ of the entries has a finite second moment, it was shown by Wigner \cite{wigner} that the empirical distribution of the eigenvalues of $X_N$, once renormalized by $\sqrt{N}$, converges almost surely and in expectation to the so-called semicircular distribution as $N$ goes to infinity. In this paper we study the same question when $P$ is in the domain of attraction of an $α$-stable law. We prove that if we renormalize the eigenvalues by a constant $a_N$ of order $N^{\frac{1}α}$, the correspo…
We consider $n\times n$ Hermitian matrices with i.i.d. entries $X_{ij}$ whose tail probabilities $\mathbb {P}(|X_{ij}|\geq t)$ behave like $e^{-at^α}$ for some $a>0$ and $α\in(0,2)$. We establish a large deviation principle for the empirical spectral measure of $X/\sqrt{n}$ with speed $n^{1+α/2}$ with a good rate function $J(μ)$ that is finite only if $μ$ is of the form $μ=μ_{\mathrm{sc}}\boxplusν$ for some probability measure $ν$ on $\mathbb {R}$, where $\boxplus$ denotes the free convolution and $μ_{\mathrm{sc}}$ is Wigner's semicircle law. We obtain explicit expressions for $J(μ_{\mathrm{sc…
We study a class of Hermitian random matrices which includes and generalizes Wigner matrices, heavy-tailed random matrices, and sparse random matrices such as the adjacency matrices of Erdos-Renyi random graphs with p ~ 1/N. Our NxN random matrices have real entries which are i.i.d. up to symmetry. The distributions may depend on N, however, the sums of rows must converge in distribution; it is then well-known that the limiting distributions are infinitely divisible. We show that a limiting empirical spectral distribution (LSD) exists, and via local weak convergence of associated graphs, the L…
The paper studies the spectral properties of large Wigner, band and sample covariance random matrices with heavy tails of the marginal distributions of matrix entries.
We present large deviations principles for the moments of the empirical spectral measure of Wigner matrices and empirical measure of $β$-ensembles in three cases : the case of Wigner matrices without Gaussian tails, that is Wigner matrices whose entries have tail distributions decreasing as $e^{-ct^α}$, for some constant $c>0$ and with $α\in (0,2)$, the case of Gaussian Wigner matrices, and the case of $β$-ensembles associated with a convex potential with polynomial growth.
We establish precise right-tail small deviation estimates for the largest eigenvalue of real symmetric and complex Hermitian matrices whose entries are independent random variables with uniformly bounded moments. The proof relies on a Green function comparison along a continuous interpolating matrix flow for a long time. Less precise estimates are also obtained in the left tail.
It is well-known that distances in random iid matrices are highly concentrated around their mean. In this note we extend this concentration phenomenon to Wigner matrices. Exponential bounds for the lower tail are also included.
Large deviations of the largest eigenvalue of supercritical sparse Wigner matrices
针对平均度至少为维度对数阶的超临界稀疏Wigner矩阵，研究其最大特征值的大偏差上尾事件，揭示由高度顶点或大权重重团引发的机制，发现速率函数在典型值处不连续，补充了常平均度情形的已有结果。
Concentration inequalities are obtained on Poisson space, for random functionals with finite or infinite variance. In particular, dimension free tail estimates and exponential integrability results are given for the Euclidean norm of vectors of independent functionals. In the finite variance case these results are applied to infinitely divisible random variables such as quadratic Wiener functionals, including Lévy's stochastic area and the square norm of Brownian paths. In the infinite variance case, various tail estimates such as stable ones are also presented.
We analyze the distribution of eigenvectors for mesoscopic, mean-field perturbations of diagonal matrices in the bulk of the spectrum. Our results apply to a generalized $N\times N$ Rosenzweig-Porter model. We prove that the eigenvectors entries are asymptotically Gaussian with a specific variance, localizing them onto a small, explicit part of the spectrum. For a well spread initial spectrum, this variance profile universally follows a heavy-tailed Cauchy distribution. In the case of smooth entries, we also obtain a strong form of quantum unique ergodicity as an overwhelming probability bound…
One-rank perturbations of Wigner matrices have been closely studied: let $P=\frac{1}{\sqrt{n}}A+θvv^T$ with $A=(a_{ij})_{1 \leq i,j \leq n} \in \mathbb{R}^{n \times n}$ symmetric, $(a_{ij})_{1 \leq i \leq j \leq n}$ i.i.d. with centered standard normal distributions, and $θ>0, v \in \mathbb{S}^{n-1}.$ It is well known $λ_1(P),$ the largest eigenvalue of $P,$ has a phase transition at $θ_0=1:$ when $θ\leq 1,$ $λ_1(P) \xrightarrow[]{a.s.} 2,$ whereas for $θ> 1,$ $λ_1(P) \xrightarrow[]{a.s.} θ+θ^{-1}.$ Under more general conditions, the limiting behavior of $λ_1(P),$ appropriately normalized, has…
We study the statistics of the largest eigenvalues of real symmetric and sample covariance matrices when the entries are heavy tailed. Extending the result obtained by Soshnikov in \cite{Sos1}, we prove that, in the absence of the fourth moment, the top eigenvalues behave, in the limit, as the largest entries of the matrix.
The well-known "Janson's inequality" gives Poisson-like upper bounds for the lower tail probability \Pr(X \le (1-\eps)\E X) when X is the sum of dependent indicator random variables of a special form. We show that, for large deviations, this inequality is optimal whenever X is approximately Poisson, i.e., when the dependencies are weak. We also present correlation-based approaches that, in certain symmetric applications, yield related conclusions when X is no longer close to Poisson. As an illustration we, e.g., consider subgraph counts in random graphs, and obtain new lower tail estimates, ex…
Tail asymptotics and precise large deviations for some Poisson cluster processes
本文研究了再生Poisson聚类过程和Hawkes过程的尾渐进行为，利用正则变化性假设和Karamata定理推导出精确的大偏差结果。
We consider the Feynman-Kac functional associated with a Brownian motion in a random potential. The potential is defined by attaching a heavy tailed positive potential around the Poisson point process. This model was first considered by Pastur (1977) and the first order term of the moment asymptotics was determined. In this paper, both moment and almost sure asymptotics are determined up to the second order. As an application, we also derive the second order asymptotics of the integrated density of states of the corresponding random Schrödinger operator.
We study heavy-tailed Hermitian random matrices that are unitarily invariant. The invariance implies that the eigenvalue and eigenvector statistics are decoupled. The motivating question has been whether a freely stable random matrix has stable eigenvalue statistics for the largest eigenvalues in the tail. We investigate this question through the use of both numerical and analytical means, the latter of which makes use of the supersymmetry method. A surprising behaviour is uncovered in that a freely stable random matrix does not necessarily yield stable statistics and if it does then it might …
Exponential Tail Estimates for Multitype Poisson Branching Processes and Application to Hawkes Processes
针对多类型泊松分支过程的总后代线性泛函，建立了显式指数矩界，并应用于多类型霍克斯过程，得到高维情形下的改进结果。
A factor matching of optimal tail between Poisson processes
针对维度$d≥3$的欧几里得空间中两个独立单位强度泊松点过程，构造了等变可测的因子完美匹配，其配对距离尾部分布以$b\textrm{exp}(-cr^d)$快速衰减，证明依赖于泊松点过程的类似尾部分配规则和分数完美匹配转完美匹配的定理。
Consider a d-dimensional Brownian motion in a random potential defined by attaching a nonnegative and polynomially decaying potential around Poisson points. We introduce a repulsive interaction between the Brownian path and the Poisson points by weighting the measure by the Feynman-Kac functional. We show that under the weighted measure, the Brownian motion tends to localize around the origin. We also determine the scaling limit of the path and also the limit shape of the random potential.
We prove that the logarithm of the determinant of a Wigner matrix satisfies a central limit theorem in the limit of large dimension. Previous results about fluctuations of such determinants required that the first four moments of the matrix entries match those of a Gaussian. Our work treats symmetric and Hermitian matrices with centered entries having the same variance and subgaussian tail. In particular, it applies to symmetric Bernoulli matrices and answers an open problem raised by Tao and Vu.
Mixed Poisson families with real-valued mixing distributions
本文证明混合泊松分布中的混合分布不必限于非负支撑，只需左尾为次指数且负值概率质量较小即可，并给出混合泊松-极值稳定族等具体例子。
Tight bounds on Poisson tails with applications to Szász-Mirakyan operators
本文为泊松尾部建立了易于处理的紧界，并基于伽马分布中位数给出简短证明，进而应用于Szász-Mirakyan算子的点态收敛速率，证明仿射函数在开区间内呈指数收敛，边界处失效。
Concentration inequalities for subgraph counts in random geometric graphs built over Poisson point processes are proved. The estimates give upper bounds for the probabilities $\mathbb{P}(N\geq M +r)$ and $\mathbb{P}(N\leq M - r)$ where $M$ is either a median or the expectation of a subgraph count $N$. The bounds for the lower tail have a fast Gaussian decay and the bounds for the upper tail satisfy an optimality condition. A special feature of the presented inequalities is that the underlying Poisson process does not need to have finite intensity measure. The tail estimates for subgraph counts…
The allocation problem for a $d$-dimensional Poisson point process is to find a way to partition the space to parts of equal size, and to assign the parts to the configuration points in a measurable, "deterministic" (equivariant) way. The goal is to make the diameter $R$ of the part assigned to a configuration point have fast decay. We present an algorithm for $d\geq3$ that achieves an $O(\operatorname {exp}(-cR^d))$ tail, which is optimal up to $c$. This improves the best previously known allocation rule, the gravitational allocation, which has an $\operatorname {exp}(-R^{1+o(1)})$ tail. The …
Heavy-tailed random variables have been used in insurance research to model both loss frequencies and loss severities, with substantially more emphasis on the latter. In the present work, we take a step toward addressing this imbalance by exploring the class of heavy-tailed frequency models formed by continuous mixtures of Negative Binomial and Poisson random variables. We begin by defining the concept of a calibrative family of mixing distributions (each member of which is identifiable from its associated Negative Binomial mixture), and show how to construct such families from only a single m…
Distributional fixed points of a Poisson shot noise transform (for nonnegative, nonincreasing response functions bounded by 1) are characterized. The tail behavior of fixed points is described. Typically they have either exponential moments or their tails are proportional to a power function, with exponent greater than -1. The uniqueness of fixed points is also discussed. Finally, it is proved that in most cases fixed points are absolutely continuous, apart from the possible atom at zero.
Poisson approximation of fixed-degree nodes in weighted random connection models
本文研究了高密度加权随机连接模型中固定度顶点的泊松近似问题，基于权重分布左尾的小权重分位数建立了一般性准则，并通过截断论证与Poisson点过程泛函的定量泊松近似结果进行证明。
In this paper, we first use the distribution of the number of records to demonstrate that the right tail probabilities of counts of rare events are generally better approximated by the right tail probabilities of Poisson distribution than {those} of normal distribution. We then show the moderate deviations in Poisson approximation generally require an adjustment and, with suitable adjustment, we establish better error estimates of the moderate deviations in Poisson approximation than those in \cite{CFS}. Our estimates contain no unspecified constants and are easy to apply. We illustrate the us…
Upper bounds for the probabilities $\mathbb{P}(F\geq \mathbb{E} F + r)$ and $\mathbb{P}(F\leq \mathbb{E} F - r)$ are proved, where $F$ is a certain component count associated with a random geometric graph built over a Poisson point process on $\mathbb{R}^d$. The bounds for the upper tail decay exponentially, and the lower tail estimates even have a Gaussian decay. For the proof of the concentration inequalities, recently developed methods based on logarithmic Sobolev inequalities are used and enhanced. A particular advantage of this approach is that the resulting inequalities even apply in set…
The Poisson Tail Conjecture for primes in short intervals
在Hardy-Littlewood猜想的条件下，研究短区间内素数计数分布的泊松尾部猜想，证明了当λ缓慢增长时计数统计与预测一致，并揭示了λ较大时的相变与偏差。
Full large deviation principles for the largest eigenvalue of sub-Gaussian Wigner matrices
研究子高斯威格纳矩阵右端最大特征值的大偏差原理，通过受限退火自由能优化问题推导出精确上尾渐进行为，改进了传统方法的假设条件。
For d>=3, we construct a non-randomized, fair and translation-equivariant allocation of Lebesgue measure to the points of a standard Poisson point process in R^d, defined by allocating to each of the Poisson points its basin of attraction with respect to the flow induced by a gravitational force field exerted by the points of the Poisson process. We prove that this allocation rule is economical in the sense that the "allocation diameter", defined as the diameter X of the basin of attraction containing the origin, is a random variable with a rapidly decaying tail. Specifically, we have the tail…
Modified ruin probability for a Cramér-Lundberg model driven by a compound mixed Poisson process
研究改进的破产概率，探讨在重尾和轻尾情况下，改进与经典破产概率的等价性及端点原子结果和密度渐近性。
We prove general nonlinear large deviation estimates similar to Chatterjee-Dembo's original bounds except that we do not require any second order smoothness. Our approach relies on convex analysis arguments and is valid for a broad class of distributions. Our results are then applied in three different setups. Our first application consists in the mean-field approximation of the partition function of the Ising model under an optimal assumption on the spectra of the adjacency matrices of the sequence of graphs. Next, we apply our general large deviation bound to investigate the large deviation …
Let Ξbe a discrete set in R^d. Call the elements of Ξcenters. The well-known Voronoi tessellation partitions R^d into polyhedral regions (of varying volumes) by allocating each site of R^d to the closest center. Here we study allocations of R^d to Ξin which each center attempts to claim a region of equal volume α. We focus on the case where Ξarises from a Poisson process of unit intensity. It was proved in math.PR/0505668 that there is a unique allocation which is stable in the sense of the Gale-Shapley marriage problem. We study the distance X from a typical site to its allocated center in th…
Universality in prelimiting tail behavior for regular subgraph counts in the Poisson regime
本文证明了在 Poisson 机制下，任意正则子图计数的上尾行为在特定阈值处发生相变，从而将此前仅在三角形情况下已知的相变普适性推广至一般正则图。
We analyze the largest eigenvalue statistics of m-dependent heavy-tailed Wigner matrices as well as the associated sample covariance matrices having entry-wise regularly varying tail distributions with parameter $0<α<4$. Our analysis extends results in the previous literature for the corresponding random matrices with independent entries above the diagonal, by allowing for m-dependence between the entries of a given matrix. We prove that the limiting point process of extreme eigenvalues is a Poisson cluster process.
We establish a large deviation principle for the empirical spectral measure of a sample covariance matrix with sub-Gaussian entries, which extends Bordenave and Caputo's result for Wigner matrices having the same type of entries [7]. To this aim, we need to establish an asymptotic freeness result for rectangular free convolution, more precisely, we give a bound in the subordination formula for information-plus-noise matrices.
The extremal tail probabilities of moving sums in a marked Poisson random field is examined here. These sums are computed by adding up the weighted occurrences of events lying within a scanning set of fixed shape and size. Change of measure and analysis of local random fields are used to provide tail probabilities. The asymptotic constants are initially expressed in a form that seems hard to evaluate and do not seem to provide any additional information on the properties of the constants. A more sophisticated approach is then undertaken giving rise to an expression that is not only neater but …
As an application of Stein's method for Poisson approximation, we prove rates of convergence for the tail probabilities of two scan statistics that have been suggested for detecting local signals in sequences of independent random variables subject to possible change-points. Our formulation deals simultaneously with ordinary and with large deviations.
In this article it is shown that if the busy period of a MGinf queue system is PME distributed, the service time is a random variable with a long tail distribution. The result is obtained through Laplace transforms analysis.
Asymptotic formulas for products of Poisson distributions
本文研究了有限个独立泊松随机变量乘积尾概率的渐进行为，利用斯特林近似、约束鞍点法和拉姆伯特函数推导出改进的拉普拉斯型渐近公式，得到具有O(log n)余项的显式近似。
We consider a spatial stochastic model of wireless cellular networks, where the base stations (BSs) are deployed according to a simple and stationary point process on $\mathbb{R}^d$, $d\ge2$. In this model, we investigate tail asymptotics of the distribution of signal-to-interference ratio (SIR), which is a key quantity in wireless communications. In the case where the path-loss function representing signal attenuation is unbounded at the origin, we derive the exact tail asymptotics of the SIR distribution under an appropriate sufficient condition. While we show that widely-used models based o…
We prove large deviation principles for two versions of fractional Poisson processes. Firstly we consider the main version which is a renewal process; we also present large deviation estimates for the ruin probabilities of an insurance model with constant premium rate, i.i.d. light tail claim sizes, and a fractional Poisson claim number process. We conclude with the alternative version where all the random variables are weighted Poisson distributed. Keywords: Mittag Leffler function; renewal process; random time cha
Deformed Fréchet law for Wigner and sample covariance matrices with tail in crossover regime
本研究拓展了变形Fréchet分布的适用范围，证明其在多种稀疏/带状Wigner矩阵及样本协方差矩阵的最大特征值中存在，同时揭示了特征值联合分布与特征向量局域化特性。
In this paper we extend the refined second-order Poincaré inequality for Poisson functionals from a one-dimensional to a multi-dimensional setting. Its proof is based on a multivariate version of the Malliavin-Stein method for normal approximation on Poisson spaces. We also present an application to partial sums of vector-valued functionals of heavy-tailed moving averages. The extension allows a functional with multivariate arguments, i.e. multiple moving averages and also multivariate values of the functional. Such a set-up has previously not been explored in the framework of stable moving av…
The Tracy-Widom distributions are among the most famous laws in probability theory, partly due to their connection with Wigner matrices. In particular, for $A=\frac{1}{\sqrt{n}}(a_{ij})_{1 \leq i,j \leq n} \in \mathbb{R}^{n \times n}$ symmetric with $(a_{ij})_{1 \leq i \leq j \leq n}$ i.i.d. standard normal, the fluctuations of its largest eigenvalue $λ_1(A)$ are asymptotically described by a real-valued Tracy-Widom distribution $TW_1:$ $n^{2/3}(λ_1(A)-2) \Rightarrow TW_1.$ As it often happens, Gaussianity can be relaxed, and this results holds when $\mathbb{E}[a_{11}]=0, \mathbb{E}[a^2_{11}]=…
Spectral aspects of random heavy-tailed tensors
研究重维格纳张量，探讨其迹不变量，证明归一化后与胖超树相关的非零渐近贡献及中心极限定理。通过分析厄多斯 - 雷尼超图等应用，得出相关矩阵经验谱分布收敛结果并推广，还推导了极限谱测度矩公式。
We consider a Wigner matrix $A$ with entries tail decaying as $x^{-α}$ with $2<α<4$ for large $x$ and study fluctuations of linear statistics $N^{-1}\operatorname{Tr}φ(A)$. The behavior of such fluctuations has been understood for both heavy-tailed matrices (i.e. $α< 2$) and light-tailed matrices (i.e. $α> 4$). This paper fills in the gap of understanding for $2<α<4$. We find that while linear spectral statistics for heavy-tailed matrices have fluctuations of order $N^{-1/2}$ and those for light-tailed matrices have fluctuations of order $N^{-1}$, the linear spectral statistics for half-heavy-…
Lévy Langevin Monte Carlo for sampling from heavy-tailed target distributions
研究如何从重尾目标分布采样，扩展Lévy朗之万蒙特卡罗方法，利用由复合泊松过程驱动的随机微分方程，可从非光滑目标和分离模式分布中采样，指数收敛到不变分布，且因复合泊松噪声项易于实现。
Sharp asymptotics for the maximal distance from the boundary to the nucleus of a typical Poisson-Voronoi cell
本文研究d维欧氏空间中典型Poisson-Voronoi胞腔顶点到核的最大距离D，推导其尾分布精确渐近，证实相关极值指数猜想并确定尾概率估计中的显式常数。
The model of heavy Wigner matrices generalizes the classical ensemble of Wigner matrices: the sub-diagonal entries are independent, identically distributed along to and out of the diagonal, and the moments its entries are of order 1/N, where N is the size of the matrices. Adjacency matrices of Erdös-Renyi sparse graphs and matrices with properly truncated heavy tailed entries are examples of heavy Wigner matrices. We consider a family X_N of independent heavy Wigner matrices and a family Y_N of arbitrary random matrices, independent of X_N, with a technical condition (e.g. the matrices of Y_N …
Let $N$ be the number of triangles in an Erdős-Rényi graph $\mathcal{G}(n,p)$ on $n$ vertices with edge density $p=d/n,$ where $d>0$ is a fixed constant. It is well known that $N$ weakly converges to the Poisson distribution with mean ${d^3}/{6}$ as $n\rightarrow \infty$. We address the upper tail problem for $N,$ namely, we investigate how fast $k$ must grow, so that the probability of $\{N\ge k\}$ is not well approximated anymore by the tail of the corresponding Poisson variable. Proving that the tail exhibits a sharp phase transition, we essentially show that the upper tail is governed by P…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。