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## Limit theorems for sums of random variables with mixture distribution

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 of the sums due to various relations between these parameters.

Keywords: Phase transitionФазовый переходcentral limit theoremцентральная предельная теоремаstable distributionsустойчивые распределенияmixture distributionсмесь распределений

Publication based on the results of:

Panov V., Limit theorems for sums of random variables with mixture distribution / 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

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

Maslov V., Математические заметки 2011 Т. 89 № 2 С. 272-284

In this paper, we construct a new distribution corresponding to a real noble gas as well as the equation of state for it. ...

Added: April 12, 2012

Bufetov A. I., Mkrtchyan S., Scherbina M. et al., Entropy and the Shannon-McMillan-Breiman theorem for beta random matrix ensembles / 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

Maslov V., Russian Journal of Mathematical Physics 2012 Vol. 19 No. 4 P. 485-499

It is shown that the Boltzmann-Maxwell distribution is not applicable to the generalized notion of ideal gas and corresponds to this ideal gas at low density only. The behavior of an ideal gas and its transition to liquid is studied as a result of a kind of generalization of the problem of “partitio numerorum”. The ...

Added: January 17, 2013

Molchanov S., Whitmeyer J., Applicable Analysis 2015

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Added: June 22, 2016

Korotayev A., Grinin L. E., World Futures: Journal of General Evolution 2009 Vol. 65 No. 7 P. 477-506

Added: March 7, 2013

49606783, Mathematical notes 2010 Vol. 88 No. 5 P. 723-731

The author constructs a one-particle distribution generalizing the Maxwell-Boltzmann and Bose-Einstein distributions for classical particles, without incurring the Gibbs paradox, and, simultaneously, constructs the thermodynamics of fluids and phase transitions on the basis of the concept of creation of dimes and clusters and obtains well-known laws. ...

Added: April 13, 2012

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

Maslov V., Mathematical notes 2012 Vol. 92 No. 3 P. 402-411

Negative pressure also means negative energy and, therefore, “holes”, antiparticles. Continuation across infinity to negative energies is accomplished by using a parastatistical correction to the Bose-Einstein distribution. ...

Added: January 17, 2013

Saakian D., Yakushkina T., Hu C., Scientific Reports 2016 Vol. 6

We propose a modification of the Crow-Kimura and Eigen models of biological molecular evolution to include a mutator gene that causes both an increase in the mutation rate and a change in the fitness landscape. This mutator effect relates to a wide range of biomedical problems. There are three possible phases: mutator phase, mixed phase ...

Added: September 16, 2016

Chugunov A., Volynsky P. E., Krylov N. et al., Scientific Reports 2014 Vol. 4 No. 7462 P. 1-8

Archaeal plasma membranes appear to be extremely durable and almost impermeable to water and ions, in contrast to the membranes of Bacteria and Eucaryota. Additionally, they remain liquid within a temperature range of 0–100° С. These are the properties that have most likely determined the evolutionary fate of Archaea, and it may be possible for bionanotechnology ...

Added: July 21, 2015

Gordeev I., Kolotova L., Starikov S., Scripta Materialia 2022 Vol. 210 Article 114481

The mechanism of Al-Si alloy crystallization from an amorphous state is still unclear. Despite the absence of equilibrium compounds for this binary system, there are several experimental evidences confirming the formation of metastable silicide at annealing of amorphous Si mixed with Al. Classification of this metastable structure is a missing puzzle piece in the description ...

Added: January 28, 2022

Yakushkina T., Saakian D., Physica A: Statistical Mechanics and its Applications 2018 Vol. 507 P. 470-477

In this paper, the impact of lethal mutations on evolutionary dynamics of asexual populations is analyzed. We suggest distinguishing different definitions of lethality, which lead to different mathematical formalizations of the microscopic model. Most of the studies focus on polyphasic lethality, meaning that individuals carrying lethal mutations have no offspring but consume common resources. In ...

Added: June 22, 2018

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

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

Pankov A., Goryainova E. R., Жерносек А. И., М. : МАИ, 2013

Данное учебное пособие предназначено для методического обеспечения цикла лабораторных работ по курсу «Теория вероятностей и математическая статистика», «Статистические методы в социологии и экономике». Описание каждой лабораторной работы содержит следующие разделы: название и цель работы; теоретические сведения; порядок выполнения работы; контрольные вопросы. Первые две лабораторные работы посвящены предварительному статистическому анализу, третья – экспериментальной проверке центральной предельной ...

Added: July 11, 2013

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

49606783, Russian Journal of Mathematical Physics 2016 Vol. 23 No. 4 P. 509-524

We show how the abstract analytic number theory of Maier, Postnikov, and others can be extended to include negative numbers and apply this to thermodynamics, information theory, and human thermodynamics. In particular, we introduce a certain large number N_0 on the "zero level" with a high multiplicity number qi ≫ 1 related to the physical ...

Added: February 25, 2017

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

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

Chetverikov V., Aminova A., Journal of Physics: Conference Series 2019 Vol. 1163 No. 012078 P. 1-6

The results of numerical calculations for the mathematical model proposed for describing the magnetization in a thin film of a ferromagnetic semiconductor at temperatures below the Curie temperature in the presence of an external electric field are presented. The theoretical prediction of the existence of a piecewise continuous solution, which describes the presence of the ...

Added: April 2, 2019

Tamm M., Nechaev S. K., Valba O. V., Journal of Statistical Mechanics: Theory and Experiment 2017 Vol. 2017 No. 053301 P. 1-17

We study the asymptotic behavior of the number of paths of length N on several classes of infinite graphs with a single special vertex. This vertex can work as an ‘entropic trap’ for the path, i.e. under certain conditions the dominant part of long paths becomes localized in the vicinity of the special point instead ...

Added: October 19, 2017

49606783, Russian Journal of Mathematical Physics 2011 Vol. 18 No. 2 P. 163-175

For a gas mixture, the new concept of number-theoretic internal energy is introduced. This energy does not depend on the masses of the miscible gases. ...

Added: April 12, 2012