• A
  • A
  • A
  • АБВ
  • АБВ
  • АБВ
  • A
  • A
  • A
  • A
  • A
Обычная версия сайта
  • RU
  • EN
  • HSE University
  • Publications
  • Articles
  • From Classical to Modern Nonlinear Central Limit Theorems
  • RU
  • EN
Расширенный поиск
Высшая школа экономики
Национальный исследовательский университет
Priority areas
  • business informatics
  • economics
  • engineering science
  • humanitarian
  • IT and mathematics
  • law
  • management
  • mathematics
  • sociology
  • state and public administration
by year
  • 2028
  • 2027
  • 2026
  • 2025
  • 2024
  • 2023
  • 2022
  • 2021
  • 2020
  • 2019
  • 2018
  • 2017
  • 2016
  • 2015
  • 2014
  • 2013
  • 2012
  • 2011
  • 2010
  • 2009
  • 2008
  • 2007
  • 2006
  • 2005
  • 2004
  • 2003
  • 2002
  • 2001
  • 2000
  • 1999
  • 1998
  • 1997
  • 1996
  • 1995
  • 1994
  • 1993
  • 1992
  • 1991
  • 1990
  • 1989
  • 1988
  • 1987
  • 1986
  • 1985
  • 1984
  • 1983
  • 1982
  • 1981
  • 1980
  • 1979
  • 1978
  • 1977
  • 1976
  • 1975
  • 1974
  • 1973
  • 1972
  • 1971
  • 1970
  • 1969
  • 1968
  • 1967
  • 1966
  • 1965
  • 1964
  • 1963
  • 1958
  • More
Subject
News
September 18, 2026
When Pictures Hinder Understanding: Illustrations May Impede Learning of Abstract Ideas
Illustrations can help remember specific actions but do not always make abstract ideas easier to learn. Researchers from HSE University and Humboldt University compared how people learn from texts with different levels of abstractness. They found that participants remembered illustrations better and performed better on related tasks after reading a multimedia text about yoga asanas than after reading an abstract text about the Nash equilibrium. The findings could help improve the selection of illustrations for educational and informational materials. The study has been published in Learning and Instruction.
September 17, 2026
'I Wish That People Would Place Greater Trust in Science'
When Tatiana Eremicheva chose Fundamental and Computational Linguistics as her field of study, she thought it would be about learning languages. Instead, she discovered it was about helping people. In this interview for the HSE Young Scientists project, she discusses science as a way of understanding the world, billiards as a team-building activity, and why learning to read is not always as easy as it seems.
September 15, 2026
Immunity to Chaos: How Personal Resources Help Us Cope with the Challenges of a Turbulent World
International conflicts, crises and digital overload—the modern world puts our minds to the test every day. Traditional psychology often focuses on the consequences: anxiety, depression, and psychosomatic disorders. But what if we looked at the problem differently—through the lens of the resources that prevent us from breaking down? Psychological immunity is precisely this set of resources. Alena Zolotareva and her group, Psychological Immunity as a Resource for Positive Functioning, are developing an integrative model of this phenomenon, adapting diagnostic tools and preparing for large-scale empirical research. Why do psychologists need to collaborate with medical professionals, and how could their research transform preventive care in clinics and corporations?

 

Have you spotted a typo?
Highlight it, click Ctrl+Enter and send us a message. Thank you for your help!

Publications
  • Books
  • Articles
  • Chapters of books
  • Working papers
  • Report a publication
  • Research at HSE

?

From Classical to Modern Nonlinear Central Limit Theorems

Mathematics. 2024. Vol. 12. No. 14. Article 2276.
Ulyanov V. V.

In 1733, de Moivre, investigating the limit distribution of the binomial distribution, was the first to discover the existence of the normal distribution and the central limit theorem (CLT). In this review article, we briefly recall the history of classical CLT and martingale CLT, and introduce new directions of CLT, namely Peng’s nonlinear CLT and Chen–Epstein’s nonlinear CLT, as well as Chen–Epstein’s nonlinear normal distribution function.

Research target: Computer Science Mathematics
Language: English
Full text
DOI
Text on another site
Keywords: central limit theoremmartingale CLTnonlinear Chen—Epstein CLTnonlinear Peng’s CLT
Publication based on the results of:
Development and theoretical analysis of new effective stochastic machine learning algorithms (2024)
Similar publications
Четырехмерные гиперэллиптические многообразия, определяемые векторными раскрасками простых многогранников
Ероховец Николай Юрьевич, Математический сборник 2026 Т. 217 № 5 С. 45–89
Toric topology assigns to each simple convex n-polytope P with m facets an n-dimensional real moment-angle manifold RZP with a canonical action of Zm2=(Z/2Z)m. We consider (not necessarily free) actions of subgroups H⊂Zm2 on RZP. The orbit space N(P,H)=RZP/H carries an action of Zm2/H. For general n we introduce the notion of Hamiltonian C(n,k)-subcomplex in the boundary of an ...
Added: September 17, 2026
Explicit bases for Riemann-Roch spaces on elliptic curves and their application in constructing various elliptic code families
Kuninets A., Malygina E., Cryptography and Communications 2026
In this paper, we determine explicit bases for Riemann–Roch spaces associated with various families of elliptic codes. We establish the feasibility and provide exact algorithms for constructing bases of Riemann–Roch spaces corresponding to arbitrary divisors on elliptic curves, including the non-effective case. These results are subsequently applied to derive bases for quasi-cyclic elliptic codes and ...
Added: September 17, 2026
Робототехника и искусственный интеллект: международные практики и возможности для России, Дальнего Востока и Арктики
Кузнецов М. Е., Полякова М., Лукьянович В. et al., ФАНУ "Востокгосплан", 2026.
Обзор международных практик развития робототехники и искусственного интеллекта и оценка возможностей их применения в условиях России, в первую очередь для Дальнего Востока и Арктической зоны РФ ...
Added: September 16, 2026
Universality of robust chaotic dynamics in a family of one-dimensional Lorenz maps
Kazakov A., Koryakin V., Safonov K. et al., Journal of Differential Equations 2026 Vol. 480 Article 114626
We study a family of one-dimensional maps that models the dynamics of a system of differential equations with a Lorenz attractor near a bifurcation curve where the system has a pair of homoclinic loops with zero separatrix value. Of particular interest is the region of the parameter plane where the map has a robust chaotic ...
Added: September 16, 2026
Continuously pleated torus or strange nonchaotic attractors seen as continuously fractalized quasiperiodic attractors
Letellier C., Stankevich N., Houri S. et al., Chaos 2026 Vol. 36 No. 9 Article 093128
Toroidal chaos designates a chaotic solution that is structured in the neighborhood of a torus. For being chaotic, such a torus needs to be discontinuously fractalized to present a Cantor set, ensuring the great sensitivity to initial conditions required for chaos. A fractal is related to self-similarity and a non-integer dimension. In fact, although clearly ...
Added: September 16, 2026
Vague stimulating ideas, images and metaphors in scientific thinking: researchers' work with horizons of unclear knowledge
Poddiakov A., / Series Social Science Research Network "Social Science Research Network". 2026. No. 7437658.
Clarity of knowledge and reasoning is necessary in many cases. Yet vagueness in scientific thinking related to surprise, curiosity, "ability to engage with not-knowing" (de Freitas) and abductive reasoning is also a crucially important source of scientific creativity which supplements combinatorial logic when dealing with the already known. Starting from studies by C. S. Peirce ...
Added: September 15, 2026
Semi-fragile image watermarking: A comprehensive survey, classification, and research directions
Melman A., Evsyutin O., Computer Science Review 2027 Vol. 63 (A) Article 101066
One of the effective means of protecting digital images is invisible watermarking. Different scenarios have different requirements for watermark resistance to post-processing. Robust watermarking schemes are resistant to a wide range of modifications. Fragile watermarking schemes are designed for integrity control and are destroyed by even minor modifications of images. Semi-fragile watermarking schemes combine features ...
Added: September 15, 2026
Автоматизированное формирование журналов событий на основе неструктурированных Интернет-источников для задач анализа процессов
Воронова К. Д., Lyadova L. N., Proceedings of the Institute for System Programming of the RAS 2026 Vol. 38 No. 4 P. 153–170
Title: Automated Event Logs Generation Based on Unstructured Internet Sources for Process Analysis Tasks Abstract. This paper presents an approach to automated structuring event-related information extracted from unstructured textual Internet sources for process mining tasks. In many practical cases, information on events associated with various processes is not presented in the form of ready-made event logs, but is ...
Added: September 14, 2026
Intelligent Decision Technologies. Proceedings of the 17th KES-IDT 2025 Conference
Springer, 2026.
This book gathers selected papers from the KES-IDT 2025 conference, held in Solin, Croatia, on June 25–27, 2025. The book presents and discusses the latest research results and generates new ideas in the field of intelligent decision-making. The range of topics discussed is classification, prediction, data analysis, big data, data science, decision support, knowledge engineering, ...
Added: September 13, 2026
Интегральная оценка сценариев кибератак на АСУ ТП с применением цифрового двойника и киберполигона
Lebedev O. B., Митяков Е. С., Саенко И. Б. et al., Информация и космос 2026 № 1 С. 53–58
The article presents a methodology for the integrated assessment of cyberattacks on industrial control systems (ICS) using a combination of the Ampire cyber range and a digital twin. The approach is based on calculating five criteria—severity, attacker capabilities, vulnerability, complexity, and defense effectiveness—that form an integrated threat index. Testing across three scenarios confirmed the index's ...
Added: September 13, 2026
Децентрализованное управление группой автономных подвижных объектов при формировании траектории движения
Lebedev O. B., Лебедев Б. К., Бесхмельнов М. И., Известия ЮФУ. Технические науки 2024 № 2(242) С. 177–190
This article examines algorithms for generating motion trajectories for unmanned aerial vehicles (UAVs) during search-and-rescue and emergency response operations. It describes methods and algorithms for controlling the movement of a UAV group in various formations—such as line, rank, column, and during turns. Control is achieved through alternative collective adaptation algorithms based on principles of collective ...
Added: September 13, 2026
Управление передвижением группы БПЛА с соблюдением геометрической структуры строя на основе альтернативной коллективной адаптации
Lebedev O. B., Котов Д. В., Известия ЮФУ. Технические науки 2024 № 1(237) С. 155–167
The primary approach to solving motion planning and control problems involves the use of intelligent technologies. These technologies are applied to define and adjust control objectives and action programs, as well as to formulate control algorithms under conditions of uncertainty—stemming from various factors—across actuator elements and the motion control, planning, and behavioral subsystems. This paper ...
Added: September 13, 2026
Гибридный алгоритм оптимизации на основе объединения моделей роевого интеллекта: роя хромосом и пчел
Lebedev O. B., Лебедев Б. К., Ганжур М. А., Информатизация и связь 2023 № 2 С. 80–84
A hybrid algorithm has been developed to solve the VLSI component placement problem by combining procedures based on bee colony behavior and a chromosome swarm. The optimization criterion is based on estimates of the number of nets crossing specific lines within the routing area. A modified chromosome swarm paradigm is described; unlike the canonical method, ...
Added: September 13, 2026
МОДЕЛИРОВАНИЕ ПОСТРОЕНИЯ ТРАЕКТОРИИ ПОДВИЖНОГО ОБЪЕКТА НА ОСНОВЕ МЕТОДОВ МНОГОКРИТЕРИАЛЬНОЙ ОПТИМИЗАЦИИ
Lebedev O. B., Лебедев Б. К., Бесхмельнов М. И., Вестник Тверского государственного технического университета 2025 № 3(27) С. 103–119
The article presents a detailed analysis of a hybrid path-planning algorithm designed for navigation in a two-dimensional space, accounting for partial uncertainty regarding terrain information. The algorithm operates in stages and is divided into two levels. At the first level, the object's local visibility zone is established, and the corresponding region on the map is ...
Added: September 12, 2026
ГИБРИДНЫЙ МЕТОД ПЛАНИРОВАНИЯ КОНФИГУРАЦИИ МАРШРУТА НА КАРТЕ МЕСТНОСТИ В УСЛОВИЯХ ЧАСТИЧНОЙ НЕОПРЕДЕЛЕННОСТИ
Lebedev O. B., Лебедев Б. К., Бесхмельнов М. И., Известия ЮФУ. Технические науки 2025 Т. 2(244) С. 68–82
This paper describes a hybrid situational trajectory planning algorithm for 2D space operating under conditions of partial uncertainty. Based on the integration of wave-propagation and ant colony optimization algorithms, it enables the real-time generation of minimum-length trajectories while simultaneously optimizing other path quality criteria. The processes of generating a trajectory segment and moving the object ...
Added: September 12, 2026
OPTIMIZATION BY THE SWARM OF TRANSFORMING CHROMOSOMES METHOD
Lebedev O. B., Beskhielnov M. I., OPTIMIZATION BY THE SWARM OF TRANSFORMING CHROMOSOMES METHOD 2024 Vol. 34 No. 3 P. 652–658
A new strategy for finding effective solutions based on modification and hybridization of the canonical representations of the particle swarm algorithm is proposed. A paradigm of a swarm of transforming chromosomes is described, making it possible to represent solutions in the affine space of chromosomes with integer parameter values. The mechanisms of chromosome transformation in ...
Added: September 12, 2026
Использование методов машинного обучения для повышения эффективности систем противодействия многоэтапных кибератак
Lebedev O. B., Левченко Д. Д., Черкасов Р. И., Инженерный вестник Дона 2026 № 2(134) Статья 7
This article analyzes the impact of artificial intelligence (AI) and machine learning technologies on the development and transformation of cyberthreats and the creation of highly effective cyberdefense systems. Key trends in AI evolution are discussed, including data-, model-, application-, and human-centric approaches, and their role in shaping both defensive and offensive capabilities. It is shown ...
Added: September 12, 2026
Характеристические алгебры и интегрируемые системы экспоненциального типа
Смирнов С. В., Миллионщиков Д. В., Уфимский математический журнал 2021 Т. 13 № 2 С. 44–73
В данной работе изучаются характеристические алгебры для систем экспоненциального типа, соответствующих вырожденным матрицам Картана. Эти системы обобщают хорошо известные в теории интегрируемых систем гиперболические уравнения синус-Гордон и Цицейки. Для таких систем, соответствующих матрицам Картана ранга 2, характеристические алгебры описаны явно в терминах образующих и соотношений, и доказано, что они имеют линейный рост. Исследуется связь между ...
Added: September 11, 2026
Darboux formulae for linear hyperbolic equations in the discrete case
Смирнов С. В., Glasgow Mathematical Journal 2026 Vol. 68 No. 2 P. 299–316
In the second half of the 19th century Darboux obtained determinant formulae that provide the general solution for a linear hyperbolic second order PDE with finite Laplace series. These formulae played an important role in his study of the theory of surfaces and, in particular, in the theory of conjugate nets. During the last three ...
Added: September 11, 2026
Integral preserving discretization of 2D Toda lattices
Смирнов С. В., Journal of Physics A: Mathematical and Theoretical 2023 Vol. 56 No. 26 Article 265204
There are different methods of discretizing integrable systems. We consider semi-discrete analog of two-dimensional Toda lattices associated to the Cartan matrices of simple Lie algebras that was proposed by Habibullin in 2011. This discretization is based on the notion of Darboux integrability. Generalized Toda lattices are known to be Darboux integrable in the continuous case ...
Added: September 11, 2026
Random walks on rank one symmetric spaces of noncompact type
Gnetov F., Konakov V., / Series arXiv "math". 2025. No. 2512.04667.
We establish a central limit theorem, a local limit theorem, and a law of large numbers for a natural random walk on a symmetric space M of non-compact type and rank one. This class of spaces, which includes the complex and quaternionic hyperbolic spaces and the Cayley hyperbolic plane, generalizes the real hyperbolic space Hn. Our approach introduces ...
Added: December 5, 2025
Multi-normex distributions for the sum of random vectors. Rates of convergence
Kratz M., Prokopenko E., Extremes 2023 Vol. 26 No. 3 P. 509–544
We build a sharp approximation of the whole distribution of the sum of iid heavy-tailed random vectors, combining mean and extreme behaviors. It extends the so-called ’normex’ approach from a univariate to a multivariate framework. We propose two possible multi-normex distributions, named d-Normex and MRV-Normex. Both rely on the Gaussian distribution for describing the mean behavior, ...
Added: February 20, 2025
Asymptotic Properties of a Statistical Estimator of the Jeffreys Divergence: The Case of Discrete Distributions
Glinskiy V., Artem Logachov, Logachova O. et al., Mathematics 2024 Vol. 12 No. 21 Article 3319
We investigate the asymptotic properties of the plug-in estimator for the Jeffreys divergence, the symmetric variant of the Kullback–Leibler (KL) divergence. This study focuses specifically on the divergence between discrete distributions. Traditionally, estimators rely on two independent samples corresponding to two distinct conditions. However, we propose a one-sample estimator where the condition results from a ...
Added: February 19, 2025
Note on normal approximation for number of triangles in heterogeneous Erdos-Renyi graph
Logachov A.V., Mogulskii A. A., Yambartsev A. A., Siberian Electronic Mathematical Reports 2024 Vol. 21 No. 2 P. 914–926
We obtain a bound for the convergence rate in the central limit theorem for the number of triangles in a heterogeneous Erdos-Renyi graphs. Our approach is reminiscent of Hoeffding decomposition (a common technique in the theory of U-statistics). We show that the centered and normalized number of triangles asymptotically behaves as the normalized sum of ...
Added: February 19, 2025
  • About
  • About
  • Key Figures & Facts
  • Sustainability at HSE University
  • Faculties & Departments
  • International Partnerships
  • Faculty & Staff
  • HSE Buildings
  • HSE University for Persons with Disabilities
  • Public Enquiries
  • Studies
  • Admissions
  • Programme Catalogue
  • Undergraduate
  • Graduate
  • Exchange Programmes
  • Summer University
  • Summer Schools
  • Semester in Moscow
  • Business Internship
  • Research
  • International Laboratories
  • Research Centres
  • Research Projects
  • Monitoring Studies
  • Conferences & Seminars
  • Academic Jobs
  • Yasin (April) International Academic Conference on Economic and Social Development
  • Media & Resources
  • Publications by staff
  • HSE Journals
  • Publishing House
  • iq.hse.ru: commentary by HSE experts
  • Library
  • Economic & Social Data Archive
  • Video
  • HSE Repository of Socio-Economic Information
  • HSE1993–2026
  • Contacts
  • Copyright
  • Privacy Policy
  • Site Map
Edit