Related papers: Some Notes on Quantitative Generalized CLTs with S…
We prove a convergence theorem for U-statistics of degree two, where the data dimension $d$ is allowed to scale with sample size $n$. We find that the limiting distribution of a U-statistic undergoes a phase transition from the…
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 $\alpha$-stable laws and…
In this paper we provide a detailed convergence analysis for an unconditionally energy stable, second-order accurate convex splitting scheme for the Modified Phase Field Crystal equation, a generalized damped wave equation for which the…
We embark on a systematic study of continuous non-invertible symmetries, focusing on 1+1d CFTs. We describe a generalized version of Noether's theorem, where continuous non-invertible symmetries are associated to $\textit{non-local}$…
In this article, we prove the convergence of a semi-discrete numerical method applied to a general class of nonlocal nonlinear wave equations where the nonlocality is introduced through the convolution operator in space. The most important…
We study the problem of approximately recovering a probability distribution given noisy measurements of its Chebyshev polynomial moments. This problem arises broadly across algorithms, statistics, and machine learning. By leveraging a…
In this paper, we address the issue on non-asymptotic convergence bounds of Euler-type schemes associated with non-dissipative SDEs. On the one hand, for non-degenerate SDEs with super-linear drifts, we propose a novel modified Euler scheme…
We consider a nonlocal approximation of the quadratic porous medium equation where the pressure is given by a convolution with a mollification kernel. It is known that when the kernel concentrates around the origin, the nonlocal equation…
We derive conditions under which random sequences of polarizations (two-point symmetrizations) converge almost surely to the symmetric decreasing rearrangement. The parameters for the polarizations are independent random variables whose…
In this paper, we give a short proof of the weak convergence to the Kesten-McKay distribution for the normalized spectral measures of random $N$-lifts. This result is derived by generalizing a formula of Friedman involving Chebyshev…
Various quantum analogues of the central limit theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It…
In this article, we establish a near-optimal convergence rate for the CLT of linear eigenvalue statistics of Wigner matrices, in Kolmogorov-Smirnov distance. For all test functions $f\in C^5(\mathbb R)$, we show that the convergence rate is…
In this paper we provide a detailed convergence analysis for fully discrete second order (in both time and space) numerical schemes for nonlocal Allen-Cahn (nAC) and nonlocal Cahn-Hilliard (nCH) equations. The unconditional unique…
We express generalized Cauchy-Stieltjes transforms of some particular Beta distributions (of ultraspherical type generating functions for orthogonal polynomials) as a powered Cauchy-Stieltjes transform of some measure. For suitable values…
We investigate the existence of steady states and exponential decay for hypocoercive Fokker--Planck equations on the whole space with drift terms that are linear in the position variable. For this class of equations, we first establish that…
Divergence functions are measures of distance or dissimilarity between probability distributions that serve various purposes in statistics and applications. We propose decompositions of Wasserstein and Cram\'er distances$-$which compare two…
This paper studies the central limit theorems (CLTs) for linear spectral statistics (LSSs) of general sample covariance matrices, when the test functions belong to $C^3$, the class of functions with continuous third order derivatives. We…
Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in…
In this paper, we establish the central limit theorem (CLT) for linear spectral statistics (LSS) of large-dimensional sample covariance matrix when the population covariance matrices are not uniformly bounded, which is a nontrivial…
This paper investigates the central limit theorem for linear spectral statistics of high dimensional sample covariance matrices of the form $\mathbf{B}_n=n^{-1}\sum_{j=1}^{n}\mathbf{Q}\mathbf{x}_j\mathbf{x}_j^{*}\mathbf{Q}^{*}$ where…