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## A multivariate central limit theorem for weighted sums

P. 21-23.

Ayvazyan S. A., Ulyanov V. V.

We consider the typical behavior of the weighted sums of independent identically distributed random vectors in k-dimensional space. It is shown that in this case the rate of convergence in the multivariate central limit theorem is of order O(1/n) up to logarithmic factor. This extends the one-dimensional Klartag and Sodin result.

Language:
English

Publication based on the results of:

Nikitin Y. Y., Petrov V. V., Zaitsev A. Y. et al., Vestnik of the St. Petersburg University: Mathematics 2018 Vol. 51 No. 2 P. 201-232

This is the first in a series of reviews devoted to the scientific achievements of the Leningrad–St. Petersburg school of probability and statistics in the period from 1947 to 2017. It is devoted to limit theorems for sums of independent random variables—a traditional subject for St. Petersburg. It refers to the classical limit theorems: the ...

Added: October 1, 2019

Nikita Puchkin, Vladimir Ulyanov, Annales de l'institut Henri Poincare (B) Probability and Statistics 2023 Vol. 59 No. 3 P. 1508-1529

We show that external randomization may enforce the convergence of test statistics to their limiting distributions in particular cases. This results in a sharper inference. Our approach is based on a central limit theorem for weighted sums. We apply our method to a family of rank-based test statistics and a family of phi-divergence test statistics ...

Added: September 3, 2023

Dragunova K., Гаращенкова А. А., Remizov I., / Cornell University. Series arXiv "math". 2021.

Chernoff approximations are a flexible and powerful tool of functional analysis, which can be used, in particular, to find numerically approximate solutions of some differential equations with variable coefficients. For many classes of equations such approximations have already been constructed, however, the speed of their convergence to the exact solution has not been properly studied. ...

Added: December 16, 2021

Carlier G., Eichinger K., Kroshnin A., SIAM Journal on Mathematical Analysis 2021 Vol. 53 No. 5 P. 5880-5914

In this paper, we investigate properties of entropy-penalized Wasserstein barycenters introduced in [J. Bigot, E. Cazelles, and N. Papadakis, SIAM J. Math. Anal., 51 (2019), pp. 2261--2285] as a regularization of Wasserstein barycenters [M. Agueh and G. Carlier, SIAM J. Math. Anal., 43 (2011), pp. 904--924]. After characterizing these barycenters in terms of a system of Monge--Ampère ...

Added: October 27, 2021

Bobkov S., Ulyanov V. V., Theory of Probability and Its Applications 2022 Vol. 66 No. 4 P. 537-549

We give a short overview of the results related to the refined forms of the central limit theorem, with a focus on independent integer-valued random variables (r.v.'s). In the independent and non-identically distributed (non-i.i.d.) case, an approximation is then developed for the distribution of the sum by means of the Chebyshev--Edgeworth correction containing the moments ...

Added: February 22, 2022

Ulyanov V. V., Bobkov S., Danshina M., / CRC 1283, Bielefeld University, Bielefeld, Germany. Series 21027 "Collaborative Research Centre 1283, Bielefeld University". 2021. No. 21027.

Convergence of order O(1/ √ n) is obtained for the distance in total variation between the Poisson distribution and the distribution of the number of fixed size cycles in generalized random graphs with random vertex weights. The weights are assumed to be independent identically distributed random variables which have a power-law distribution. The proof is ...

Added: March 29, 2021

Shatskikh S. Y., Мелкумова Л. Э., , in : CEUR Workshop Proceedings. Vol. 1638: ITNT 2016, Information Technology and Nanotechnology.: CEUR-WS.org, 2016. P. 763-768.

The article is devoted to normality assumption in statistical data analysis. It gives a short historical review of the development of scientific views on the normal law and its applications. It also briefly covers normality tests and analyzes possible consequences of using the normality assumption incorrectly. ...

Added: June 8, 2021

Bufetov A. I., Mkrtchyan S., Scherbina M. et al., / Cornell University. Series math "arxiv.org". 2013. No. 1301.0342.

We show that beta ensembles in Random Matrix Theory with generic real analytic potential have the asymptotic equipartition property. In addition, we prove a Central Limit Theorem for the density of the eigenvalues of these ensembles. ...

Added: February 21, 2013

Goetze F., Naumov A.A., Tikhomirov A., Bernoulli: a journal of mathematical statistics and probability 2018 Vol. 24 No. 3 P. 2358-2400

We consider a random symmetric matrix X=[X_{jk}]_{j,k=1}^n with upper triangular entries being i.i.d. random variables with mean zero and unit variance. We additionally suppose that \E|X_{11}|^{4+\delta}=:\mu_{4+\delta}<\infty for some \deta>0. The aim of this paper is to significantly extend a recent result of the authors Götze, Naumov and Tikhomirov (2015) and show that with high probability the typical ...

Added: February 13, 2018

Gribkova N., Probability and Mathematical Statistics 2017 Vol. 37 No. 1 P. 101-118

In this paper, we propose a new approach to the investigation of asymptotic properties of trimmed L-statistics and we apply it to the Cramér type large deviation problem. Our results can be compared with those in Callaert et al. (1982) – the first and, as far as we know, the single article where some results ...

Added: February 28, 2020

Kroshnin A., Spokoiny V., Suvorikova A., Annals of Applied Probability 2021 Vol. 31 No. 3 P. 1264-1298

n this work we introduce the concept of Bures-Wasserstein barycenter $Q_*$, that is essentially a Fr\'echet mean of some distribution $P$ supported on a subspace of positive semi-definite Hermitian operators $\mathbb{H}_{+}(d)$.
We allow a barycenter to be constrained to some affine subspace of $\mathbb{H}_{+}(d)$ and provide conditions ensuring its existence and uniqueness.
We also investigate convergence and concentration properties ...

Added: October 30, 2020

Shchegolev A., / Cornell University. Серия math "arxiv.org". 2021.

In the paper, we study a new rate of convergence estimate for homogeneous discrete-time nonlinear Markov chains based on the Markov-Dobrushin condition. This result generalizes the convergence estimates for any positive number of transition steps. An example of a class such a process provided indicates that such types of estimates considering several transition steps may ...

Added: October 22, 2021

Shchegolev A., Управление большими системами: сборник трудов 2023 № 102 С. 5-14

The class of nonlinear Markov processes is characterized by the dependence of the current state of the process on its current distribution in addition to the dependence on the previous state. Due to this feature, these processes are characterized by complex limit behavior and ergodic properties, for which the usual criteria for Markov processes are ...

Added: June 12, 2023

Bobkov S., , in : Geometric Aspects of Functional Analysis. Vol. 1: Israel Seminar (GAFA) 2017-2019.: Springer, 2020. Ch. 5. P. 71-97.

We consider rates of approximation of distributions of weighted sums of independent, identically distributed random variables by the Edgeworth correction of the 4-th order. ...

Added: August 2, 2020

Panov V., / Cornell University. Series arXiv "math". 2017. No. 1703.10463.

In this paper, we study the fluctuations of sums of random variables with distribution defined as a mixture of light-tail and truncated heavy-tail distributions. We focus on the case when both the mixing coefficient and the truncation level depend on the number of summands. The aim of this research is to characterize the limiting distributions ...

Added: March 31, 2017

Sherstinova T., Martynenko G., , in : Digital Transformation and Global Society. Fourth International Conference, DTGS 2019, St. Petersburg, Russia, June 19–21, 2019, Revised Selected Papers. : Springer, 2019. P. 719-731.

Digital technologies provide new possibilities for studying cultural heritage. Thus, literature research involving large text corpora allows to set and solve theoretical problems which previously had no prospects for their decision. For example, it has become possible to model the literary system for some defi-nite literary period (i.e., for the Silver Age of Russian literature) ...

Added: October 31, 2019

Molchanov S., Panov V., Stochastics-An International Journal of Probability and Stochastic Processes 2019 Vol. 91 No. 5 P. 754-772

In this paper, we consider limit laws for the model, which is a generalisation of the random energy model (REM) to the case when the energy levels have the mixture distribution. More precisely, the distribution of the energy levels is assumed to be a mixture of two normal distributions, one of which is standard normal, ...

Added: November 19, 2018

Shchegolev A., Управление большими системами: сборник трудов 2021 № 90 С. 36-48

The paper studies an improved estimate for the rate of convergence for nonlinear homogeneous discrete-time Markov chains. These processes are nonlinear in terms of the distribution law. Hence, the transition kernels are dependent on the current probability distributions of the process apart from being dependent on the current state. Such processes often act as limits ...

Added: April 21, 2021

Vedenin A., Журнал Средневолжского математического общества 2022 Т. 24 № 3 С. 280-288

This paper is devoted to a new method for constructing approximations to the solution of a parabolic partial differential equation. The Cauchy problem for the heat equation on a straight line with a variable heat conduction coefficient is considered. In this paper, a sequence of functions is constructed that converges to the solution of the ...

Added: May 18, 2023

Goetze F., Naumov A.A., Tikhomirov A., Bernoulli: a journal of mathematical statistics and probability 2017 Vol. 23 No. 4B P. 3067-3113

In this paper we consider the product of two independent random matrices X^(1) and X^(2). Assume that X_{jk}^{(q)},1\le j,k \le n,q=1,2,, are i.i.d. random variables with \EX_{jk}^{q}=0, VarX_{jk}^{(q)}=1/ Denote by s_1(W),…,s_n(W) the singular values of W:=n^{-1}X^(1)X^(2). We prove the central limit theorem for linear statistics of the squared singular values s_1^2(W),…,s_n^2(W) showing that the limiting variance depends on \kappa_4:=\E(X_{11}^{(1)})^4−3. ...

Added: April 28, 2018

Bressaud X., Bufetov A.I., Hubert P., Proceedings of the London Mathematical Society 2014 Vol. 109 No. 2 P. 483-522

Deviation of ergodic sums is studied for substitution dynamical systems with a matrix that admits eigenvalues of modulus 1. The functions γ we consider are the corresponding eigenfunctions. In Theorem 1.1, we prove that the limit inferior of the ergodic sums (n,γ(x_0)+⋯+γ(x_{n−1})) n∈N is bounded for every point x in the phase space. In Theorem ...

Added: October 23, 2014

Molchanov S., Whitmeyer J., Applicable Analysis 2015

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Added: June 22, 2016

Ulyanov V. V., Ayvazyan S., Springer Proceedings in Mathematics & Statistics 2023 No. 425 P. 225-257

The “typical” asymptotic behavior of the weighted sums of independent random vectors in k-dimensional space is considered. It is shown that in this case the rate of convergence in the multivariate central limit theorem is of order O(1/n). This extends the one-dimensional Klartag and Sodin (2011) result ...

Added: December 1, 2023

Sawada T., Frontiers in Psychology 2021 Vol. 12 Article 762418

This study describes how the conditions in the Central Limit theorem (CLT) are usually not satisfied in empirical Psychological studies by comparing the formulation of the CLT with a common experimental procedure used in empirical Psychological studies. This explains why the CLT cannot assure that the population follows a normal distribution no matter how large ...

Added: November 9, 2021