Searcharxiv⌕ Search

arXiv subjects

G. P. Chistyakov

Publications and source records attributed to G. P. Chistyakov.

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↗
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↗
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↗
arXiv2013

The Entropic Erdős-Kac Limit Theorem

We prove entropic and total variation versions of the Erdős-Kac limit theorem for the maximum of the partial sums of i.i.d. random variables with densities.

math.PR↗
arXiv2012

Fisher information and the central limit theorem

An Edgeworth-type expansion is established for the relative Fisher information distance to the class of normal distributions of sums of i.i.d. random variables, satisfying moment conditions. The validity of the central limit theorem is studied via properties of the Fisher information along convolutions.

math.PR↗
arXiv2011

Asymptotic Expansions in the CLT in Free Probability

We prove Edgeworth type expansions for distribution functions of sums of free random variables under minimal moment conditions. The proofs are based on the analytic definition of free convolution. We apply these results to the expansion of densities to derive expansions for the free entropic distance of sums to the Wigner law.

math.PR↗
arXiv2011

Rate of Convergence in the entropic free CLT

We prove an expansion for densities in the free CLT and apply this result to an expansion in the entropic free central limit theorem assuming a moment condition of order 8 for the free summands.

math.PR↗
arXiv2011

Characterization problems for linear forms with free summands

Let $T_1,...,T_n$ denote free random variables. For two linear forms $L_1=\sum_{j=1}^n a_jT_j$ and $L_2=\sum_{j=1}^n b_jT_j$ with real coefficients $a_j$ and $b_j$ we shall describe all distributions of $T_1,...,T_n$ such that $L_1$ and $L_2$ are free. For identically distributed free random variables $T_1,...,T_n$ with distribution $μ$ we establish necessary and sufficient conditions on the coefficients $a_j,b_j,\,j=1,...,n,$ such that the statements:\quad $(i)$ $μ$ is a centered semicircular distribution; and $(ii)$ \, $L_1$ and $L_2$ are identically distributed ($L_1\stackrel{D}{=}L_2$); are equivalent. In the proof we give a complete characterization of all sequences of free cumulants of measures with compact support and with a finite number of non zero entries. The characterization is based on topological properties of regions defined by means of the Voiculescu transform $ϕ$ of such sequences.

math.PR↗