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F. Götze

Publications and source records attributed to F. Götze.

At least 19 recordsLinked to original sources
arXiv2025

Rényi Divergences in Central Limit Theorems: Old and New

We give an overview of various results and methods related to information-theoretic distances of Rényi type in the light of their applications to the central limit theorem (CLT). The first part (Sections 1-9) is devoted to the total variation and the Kullback-Leibler distance (relative entropy). In the second part (Sections 10-15) we discuss general properties of Rényi and Tsallis divergences of order $α>1$, and then in the third part (Sections 16-21) we turn to the CLT and non-uniform local limit theorems with respect to these strong distances. In the fourth part (Sections 22-31), we discuss recent results on strictly subgaussian distributions and describe necessary and sufficient conditions which ensure the validity of the CLT with respect to the Rényi divergence of infinite order.

cs.IT↗
arXiv2023

Asymptotic Expansions and two-sided Bounds in Randomized Central Limit Theorems

Lower and upper bounds are explored for the uniform (Kolmogorov) and $L^2$-distances between the distributions of weighted sums of dependent summands and the normal law. The results are illustrated for several classes of random variables whose joint distributions are supported on Euclidean spheres. We also survey several results on improved rates of normal approximation in randomized central limit theorems.

math.PR↗
arXiv2023

Strictly subgaussian probability distributions

We explore the class of probability distributions on the real line whose Laplace transform admits a strong upper bound of subgaussian type. Using Hadamard's factorization theorem, we extend the class $\mathfrak L$ of Newman and propose new sufficient conditions for this property in terms of location of zeros of the associated characteristic functions in the complex plane. The second part of this note deals with Laplace transforms of strictly subgaussian distributions with periodic components. This subclass contains interesting examples, for which the central limit theorem with respect to the Rényi entropy divergence of infinite order holds.

math.PR↗
arXiv2022

On the Largest and the Smallest Singular Value of Sparse Rectangular Random Matrices

We derive estimates for the largest and smallest singular values of sparse rectangular $N\times n$ random matrices, assuming $\lim_{N,n\to\infty}\frac nN=y\in(0,1)$. We consider a model with sparsity parameter $p_N$ such that $Np_N\sim \log^{α}N$ for some $α>1$, and assume that the moments of the matrix elements satisfy the condition $\mathbf E|X_{jk}|^{4+δ}\le C<\infty$. We assume also that the entries of matrices we consider are truncated at the level $(Np_N)^{\frac12-\varkappa}$ with $\varkappa:=\fracδ{2(4+δ)}$.

math.PR↗
arXiv2022

Local Laws for Sparse Sample Covariance Matrices without the truncation condition

We consider sparse sample covariance matrices $\frac1{np_n}\mathbf X\mathbf X^*$, where $\mathbf X$ is a sparse matrix of order $n\times m$ with the sparse probability $p_n$. We prove the local Marchenko--Pastur law in some complex domain assuming that $np_n>\log^βn$, $β>0$ and some $(4+δ)$-moment condition is fulfilled, $δ>0$.

math.PR↗
arXiv2020

Poincaré Inequalities and Normal Approximation for Weighted Sums

Under Poincaré-type conditions, upper bounds are explored for the Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law. Based on improved concentration inequalities on high-dimensional Euclidean spheres, the results extend and refine previous results to non-symmetric models.

math.PR↗
arXiv2019

Normal Approximation for Weighted Sums under a Second Order Correlation Condition

Under correlation-type conditions, we derive an upper bound of order $(\log n)/n$ for the average Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law. The result is based on improved concentration inequalities on high-dimensional Euclidean spheres. Applications are illustrated on the example of log-concave probability measures.

math.PR↗
arXiv2017

On distribution of points with conjugate algebraic integer coordinates close to planar curves

Let $φ:\mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable function on an interval $J\subset\mathbb{R}$ and let $\boldsymbolα=(α_1,α_2)$ be a point with algebraic conjugate integer coordinates of degree $\leq n$ and of height $\leq Q$. Denote by $\tilde{M}^n_φ(Q,γ, J)$ the set of points $\boldsymbolα$ such that $|φ(α_1)-α_2|\leq c_1 Q^{-γ}$. In this paper we show that for a real $0<γ<1$ and any sufficiently large $Q$ there exist positive values $c_2<c_3$, which are independent of $Q$, such that $c_2\cdot Q^{n-γ}<# \tilde{M}^n_φ(Q,γ, J)< c_3\cdot Q^{n-γ}$.

math.NT↗
arXiv2016

On points with algebraically conjugate coordinates close to smooth curves

We show that for any sufficiently large integer $Q$ and a real $0\leqλ\leq\frac34$ there exists a value $c(n,f,J)>0$ such that all strips $L(Q,λ)=\{(x,y):|y-f(x)|<Q^{-λ}, x\in J=[a,b]\}$ contain at least $c(n, f, J)Q^{n+1-λ}$ points $\barγ=(α,β)$ with algebraically conjugate coordinates. We consider points $\barγ$ such that the minimal polynomial $P(x)$ of $α,β$ is of degree $°P\leq n,\ n\ge 2$, and height $H(P)\leq Q$. The proof is based on a metric theorem on the measure of the set of vectors $(x,y)$ lying in a rectangle $Π$ of dimensions $Q^{-s_1}\times Q^{-s_2}$ with $|P(x)|, |P(y)|$ bounded from above and $|P'(x)|,|P'(y)|$ bounded from below, where $P(x)$ is a polynomial of degree $°P\leq n$ and height $H(P)\leq Q$. This theorem is a generalization of a result obtained by V. Bernik, F. Götze and O. Kukso for $s_1=s_2=\frac12$ and $λ= \frac12$.

math.NT↗
arXiv2016

Rényi divergence and the central limit theorem

We explore properties of the $χ^2$ and more general Rényi (Tsallis) distances to the normal law. In particular we provide necessary and sufficient conditions for the convergence to the normal law in the central limit theorem using these distances. Moreover, we derive exact rates of convergence in these distances with respect to an increasing number of summands.

math.PR↗
arXiv2015

Optimal bounds for convergence of expected spectral distributions to the semi-circular law for the $4+ε$ moment ensemble

This paper extends a previous bound of order $O(n^{-1})$ of the authors (arXiv:1405.7820[math.PR]), for the rate of convergence in Kolmogorov distance of the expected spectral distribution of a Wigner random matrix ensemble to the semicircular law. Here we relax the moment conditions for entries of the Wigner matrices from order $8$ to order $4+ ε$ for an arbitrary small $ε>0$.

math.PR↗
arXiv2015

Optimal Bounds for Convergence of Expected Spectral Distributions to the Semi-Circular Law

Let $\mathbf X=(X_{jk})_{j,k=1}^n$ denote a Hermitian random matrix with entries $X_{jk}$, which are independent for $1\le j\le k\le n$. We consider the rate of convergence of the empirical spectral distribution function of the matrix $\mathbf X$ to the semi-circular law assuming that ${\mathbf E} X_{jk}=0$, ${\mathbf E} X_{jk}^2=1$ and that $$ \sup_{n\ge1}\sup_{1\le j,k\le n}{\mathbf E}|X_{jk}|^4=:μ_4<\infty \quad \text{and} \sup_{1\le j,k\le n}|X_{jk}|\le D_0n^{\frac14}. $$ By means of a recursion argument it is shown that the Kolmogorov distance between the expected spectral distribution of the Wigner matrix $\mathbf W=\frac1{\sqrt n}\mathbf X$ and the semicircular law is of order $O(n^{-1})$.

math.PR↗