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Entropy and the Shannon-McMillan-Breiman theorem for beta random matrix ensembles
Cornell University
,
2013.
No. 1301.0342.
Bufetov A. I., Mkrtchyan S., Scherbina M., Soshnikov A.
Bufetov A., Journal of Mathematical Physics 2013 Vol. 54 No. 113302 P. 1-10
To a $N \times N$ real symmetric matrix Kerov assigns a piecewise linear function whose local minima are the eigenvalues of this matrix and whose local maxima are the eigenvalues of its $(N-1) \times (N-1)$ submatrix. We study the scaling limit of Kerov's piecewise linear functions for Wigner and Wishart matrices. For Wigner matrices the ...
Added: October 11, 2013
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
Bufetov A. I., Mkrtchyan S., Scherbina M. et al., Journal of Statistical Physics 2013 Vol. 152 No. 1 P. 1-14
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: March 13, 2014
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
Tolmachov K., / Cornell University. Series arXiv:1202.1806 "ArXiv.org". 2012.
In this paper we compute the precise asymptotics of the variance of linear statistic of descents on a growing interval for Plancherel Young diagrams (following Vershik and Kerov, diagrams are considered rotated by π/4). We also give an example of a local configuration with linearly growing variance in a fixed regime and prove the central limit ...
Added: December 8, 2013
Panov V., Statistics and Probability Letters 2017 Vol. 129 P. 379-386
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: July 10, 2017
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
Molchanov S., Whitmeyer J., Applicable Analysis 2015
...
Added: June 22, 2016
Natanzon S. M., Orlov A. Y., Theoretical and Mathematical Physics 2020 Vol. 204 No. 3 P. 1166-1194
To obtain a generating function of the most general form for Hurwitz numbers with arbitrary base surfaceand arbitrary ramification profiles, we consider a matrix model constructed according to a graph on anoriented connected surfaceΣwith no boundary. The vertices of this graph, called stars, are small discs,and the graph itself is a clean dessin d’enfants. We ...
Added: September 27, 2020
Chebotarev A., Долгопрудный : Физтех-полиграф, 2010
Предисловие.
Теория вероятностей возникла в XVI--XVII веках как раздел математики, объясняющий причины выигрыша или проигрыша в азартных играх. Участие знаменитых ученых потребовалось для анализа игровых стратегий и объяснения ряда фактов отнюдь не очевидных с точки зрения здравого смысла. Вероятностные методы описания окружающей реальности остаются актуальными и сейчас, более того, сложность рассматриваемых систем достигла планетарного масштаба. Стохастические ...
Added: December 28, 2013
Кибзун А., Goryainova E. R., Наумов А. В., М. : Физматлит, 2014
Учебник предназначен для начального ознакомления с основами теории вероятностей и математической статистики и развития навыков решения практических задач.
Большое внимание уделено краткости изложения полного курса «Теории вероятностей и математической статистики», состоящего из теоретического и практического материала. Структура изложения максимально приближена к лекционным и практическим занятиям. Книга может одновременно играть роль учебника, задачника и справочника.
Учебник предназначен для ...
Added: December 18, 2014
Гётце Ф., Naumov A., Tikhomirov A., Random Matrices-Theory and Applications 2020 Vol. 9 No. 4 P. 2150004
We consider products of independent \(n \times n\) non-Hermitian random matrices \(\X^{(1)}, \ldots, \X^{(m)}\). Assume that their entries, \(X_{jk}^{(q)}, 1 \le j,k \le n, q = 1, \ldots, m\), are independent identically distributed random variables with zero mean, unit variance. G\"otze -- Tikhomirov~\cite{GotTikh2011} and O'Rourke--Sochnikov~\cite{Soshnikov2011} proved that under these assumptions the empirical spectral distribution (ESD) ...
Added: September 13, 2019
Marshakov A., Миронов А. Д., Морозов А. Ю., Journal of Geometry and Physics 2011 Vol. 61 P. 1203-1222
We present a summary of current knowledge about the AGT relations for conformal blocks with additional insertion of the simplest degenerate operator, and a special choice of the corresponding intermediate dimension, when the conformal blocks satisfy hypergeometric-type differential equations in position of the degenerate operator. A special attention is devoted to representation of conformal block ...
Added: February 28, 2013
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
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
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
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
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
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
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
Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189
The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...
Added: January 28, 2020
Borzykh D., ЛЕНАНД, 2021
Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...
Added: February 20, 2021
В. Л. Попов, Математические заметки 2017 Т. 102 № 1 С. 72-80
Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...
Added: May 3, 2017
Красноярск : ИВМ СО РАН, 2013
Труды Пятой Международной конференции «Системный анализ и информационные технологии» САИТ-2013 (19–25 сентября 2013 г., г.Красноярск, Россия): ...
Added: November 18, 2013