We consider the Erdos-Renyi random graph G(n,p) inside the critical window, where p = 1/n + lambda * n^{-4/3} for some lambda in R. We proved in a previous paper (arXiv:0903.4730) that considering the connected components of G(n,p) as a sequence of metric spaces with the graph distance rescaled by n^{-1/3} and letting n go to infinity yields a non-trivial sequence of limit metric spaces C = (C_1, C_2, ...). These limit metric spaces can be constructed from certain random real trees with vertex-identifications. For a single such metric space, we give here two equivalent constructions, both of which are in terms of more standard probabilistic objects. The first is a global construction using Dirichlet random variables and Aldous' Brownian continuum random tree. The second is a recursive construction from an inhomogeneous Poisson point process on R_+. These constructions allow us to characterize the distributions of the masses and lengths in the constituent parts of a limit component when it is decomposed according to its cycle structure. In particular, this strengthens results of Luczak, Pittel and Wierman by providing precise distributional convergence for the lengths of paths between kernel vertices and the length of a shortest cycle, within any fixed limit component.
The product of two zero mean correlated normal random variables, and more generally the sum of independent copies of such random variables, has received much attention in the statistics literature and appears in many application areas. However, many important distributional properties are yet to be recorded. This review paper fills this gap by providing the basic distributional theory for the sum of independent copies of the product of two zero mean correlated normal random variables. Properties covered include probability and cumulative distribution functions, generating functions, moments an…
It has been observed that an interesting class of non-Gaussian stationary processes is obtained when in the harmonics of a signal with random amplitudes and phases, frequencies can also vary randomly. In the resulting models, the statistical distribution of frequencies determines the process spectrum while the distribution of amplitudes governs the process distributional properties. Since decoupling the distributional and spectral properties can be advantageous in applications, we thoroughly investigate a variety of properties exhibited by these models. We extend previous work that represented…
We consider fixed-point equations for probability distributions on isometry classes of measured metric spaces. The construction is required to be recursive and tree-like, but we allow loops for the geodesics between points in the support of the measure: one can think of a $β$-stable looptree decomposed around one loop. We study existence and uniqueness of solutions together with the attractiveness of the fixed-points by iterating. We obtain bounds on the Hausdorff and upper Minkowski dimension, which appear to be tight for the studied models. This setup applies to formerly studied structures a…
We consider a nonlinear random walk which, in each time step, is free to choose its own transition probability within a neighborhood (w.r.t. Wasserstein distance) of the transition probability of a fixed Lévy process. In analogy to the classical framework we show that, when passing from discrete to continuous time via a scaling limit, this nonlinear random walk gives rise to a nonlinear semigroup. We explicitly compute the generator of this semigroup and corresponding PDE as a perturbation of the generator of the initial Lévy process.
We construct local solutions to the Yang-Mills heat flow (in the DeTurck gauge) for a certain class of random distributional initial data, which includes the 3D Gaussian free field. The main idea, which goes back to work of Bourgain as well as work of Da Prato-Debussche, is to decompose the solution into a rougher linear part and a smoother nonlinear part, and to control the latter by probabilistic arguments. In a companion work, we use the main results of this paper to propose a way towards the construction of 3D Yang-Mills measures.
Existence of distributional solutions of a modified Surface Quasi-Geostrophic equation (mSQG) is proven for $μ$-almost every initial condition, where $μ$ is a suitable Gaussian measure. The result is the by-product of existence of a stationary solution with white noise marginal. This solution is constructed as a limit of random point vortices, uniformly distributed and rescaled according to the Central Limit Theorem.
Let $\{X_{n}, n\ge 1\}$ be a sequence of independent random variables with common general error distribution $GED(v)$ with shape parameter $v>0$, and let $M_{n,r}$ denote the $r$th largest order statistics of $X_{1}, X_{2}, \cdots, X_{n}$. With different normalizing constants the distributional expansions of normalized powered order statistics $|M_{n,r}|^{p}$ are established, from which the convergence rates of powered order statistics to their limits are derived. This paper generalized Hall's results on powered-extremes of normal sequence.
For a unimodular random graph $(G,ρ)$, we consider deformations of its intrinsic path metric by a (random) weighting of its vertices. This leads to the notion of the conformal growth exponent of $(G,ρ)$, which is the best asymptotic degree of volume growth of balls that can be achieved by such a reweighting. Under moment conditions on the degree of the root, we show that the conformal growth exponent of a unimodular random graph bounds its almost sure spectral dimension. This has interesting consequences for many low-dimensional models. The consequences in dimension two are particularly strong…
This paper is devoted to the asymptotic analysis of the amnesic elephant random walk (AERW) using a martingale approach. More precisely, our analysis relies on asymptotic results for multidimensional martingales with matrix normalization. In the diffusive and critical regimes, we establish the almost sure convergence and the quadratic strong law for the position of the AERW. The law of iterated logarithm is given in the critical regime. The distributional convergences of the AERW to Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as w…
This paper is devoted to the asymptotic analysis of the reinforced elephant random walk (RERW) using a martingale approach. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratic strong law for the RERW. The distributional convergences of the RERW to some Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the RERW. All our analysis relies on asymptotic results for multi-dimensional martingales with matrix normalization.
A study of the limiting behavior of delayed random sums under non-identical distributions setup
本文考虑S_{n+an} -Sn型延迟和，其中a_n是满足一定条件的可能正整数值随机变量，S_n是{G_1，G_2}中具有分布函数F_n的独立随机变量X_n的和.研究了时滞和的极限性态，证明了Chover型重对数律.这些结果推广了Vasudeva and Divanji（1992）和Chen（2008）的结果。
Phase Transition in Limiting Distributions of Coherence of High-Dimensional Random Matrices
随机矩阵的相干性（定义为随机矩阵列之间Pearson相关系数的最大值）是高维统计和信号处理等广泛应用的重要量。受这些应用的启发，本文研究了$n\times p$随机矩阵的相干性的极限律，并在全维范围内研究了$p$，特别是在超高维情况下.假设随机矩阵的列是具有共同球形分布的独立随机向量，我们给出了相干极限分布行为的完整刻画。更具体地说，相干性的极限分布分别推导出三个制度：$\frac{1}{n}\log p \to 0$，$\frac{1}{n}\log p \to β\in (0, \infty)$，和$\frac{1}{n}\log p \to\infty$。结果表明，随着$n$维数的增加，相干性的极限行为在不同的区域有很大的不同，并且表现出有趣的相变现象.应用统计和压缩感知在超高维设置也进行了讨论。
We explore the relationship between recursive distributional equations and convergence results for finite difference schemes of parabolic partial differential equations (PDEs). We focus on a family of random processes called symmetric cooperative motions, which generalize the symmetric simple random walk and the symmetric hipster random walk introduced in [Addario-Berry, Cairns, Devroye, Kerriou and Mitchell, arXiv:1909.07367]. We obtain a distributional convergence result for symmetric cooperative motions and, along the way, obtain a novel proof of the Bernoulli central limit theorem. In addi…
We present an overview of the representation theoretic techniques used to study the mixing times of random walks on finite groups. We focus on the card shuffle studied by Diaconis and Shahshahani in the 1980s and a recent improvement on their result by Teyssier
We demonstrate the convergence of the characteristic polynomial of several random matrix ensembles to a limiting universal function, at the microscopic scale. The random matrix ensembles we treat are classical compact groups and the Gaussian Unitary Ensemble. In fact, the result is the by-product of a general limit theorem for the convergence of random entire functions whose zeros present a simple regularity property.
Semiclassical analysis for Yang--Mills random connections on compact surfaces
IRMA, Université de Strasbourg（斯特拉斯堡大学）；Institut Universitaire de France（法国高等研究院）；Sorbonne Université（索邦大学）；Université Paris Cité（巴黎西岱大学）；CNRS, Laboratoire de Probabilités, Statistique et Modélisation, LPSM（法国国家科学研究中心，概率、统计与建模实验室）
