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Mathematical inequality explaining concentration of random variables
In probability theory, concentration inequalities provide mathematical bounds on the probability of a random variable deviating from some value (typically
Concentration_inequality
Spread of wealth in a society
various members or groups in a society. It shows one aspect of economic inequality or economic heterogeneity. The distribution of wealth differs from the
Distribution_of_wealth
Isoperimetric-type inequality on product spaces
probability theory field of mathematics, Talagrand's concentration inequality is an isoperimetric-type inequality for product probability spaces. It was first
Talagrand's concentration inequality
Talagrand's_concentration_inequality
Probabilistic inequality applying on sum of bounded random variables
Hoeffding's inequality was proven by Wassily Hoeffding in 1963. Hoeffding's inequality is a special case of the Azuma–Hoeffding inequality and McDiarmid's
Hoeffding's_inequality
mathematical inequalities. Agmon's inequality Askey–Gasper inequality Babenko–Beckner inequality Bernoulli's inequality Bernstein's inequality (mathematical
List_of_inequalities
Geometric inequality or concentration inequality in mathematics and probability theory
Loomis–Whitney inequality and Hölder's inequality. The second is a result of probability theory which gives a concentration inequality for log-concave
Brascamp–Lieb_inequality
Bound on probability of a random variable being far from its mean
In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) provides an upper bound on the probability of deviation
Chebyshev's_inequality
developed nations. However, the inequality indicators vary considerably from state to state. States that have a high concentration of skilled jobs, implement
List of U.S. states and territories by income inequality
List_of_U.S._states_and_territories_by_income_inequality
Inequality in probability theorem
Cantelli's inequality (also called the Chebyshev-Cantelli inequality and the one-sided Chebyshev inequality) is an improved version of Chebyshev's inequality for
Cantelli's_inequality
Integral inequality
Prékopa–Leindler inequality is an integral inequality closely related to the reverse Young's inequality, the Brunn–Minkowski inequality and a number of
Prékopa–Leindler_inequality
Inequalities in probability theory
Bernstein inequalities are also known as the Chernoff bound, Hoeffding's inequality and Azuma's inequality. The martingale case of the Bernstein inequality is
Bernstein inequalities (probability theory)
Bernstein_inequalities_(probability_theory)
Probability equation in mathematics
using Hölder's inequality in place of the Cauchy-Schwarz inequality. Cantelli's inequality Second moment method Concentration inequality – a summary of
Paley–Zygmund_inequality
Statistical parameter
{\displaystyle \sigma _{n}(S^{n})=1} ), one can deduce the following concentration inequality: σ n ( A ε ) ≥ 1 − C exp ( − c n ε 2 ) , {\displaystyle \sigma
Concentration_of_measure
French mathematician (born 1952)
new aspects of the isoperimetric and concentration of measure phenomena for product spaces, by obtaining inequalities which make use of new kinds of distances
Michel_Talagrand
Concept in probability theory
In probability theory, Markov's inequality gives an upper bound on the probability that a non-negative random variable is greater than or equal to some
Markov's_inequality
Trust meant to last for multiple generations
impacts. Dynasty trusts have become a focal point in debates about wealth inequality and economic mobility. Critics argue that by enabling vast fortunes to
Dynasty_trust
Exponentially decreasing bounds on tail distributions of random variables
inequalities Concentration inequality − a summary of tail-bounds on random variables. Cramér's theorem Entropic value at risk Hoeffding's inequality Matrix
Chernoff_bound
Probability and computer science concept
theoretical computer science, McDiarmid's inequality (named after Colin McDiarmid ) is a concentration inequality which bounds the deviation between the
McDiarmid's_inequality
Probabilistic inequality
concerns distance from the mean without requiring unimodality Concentration inequality – a summary of tail-bounds on random variables. Winkler A. (1886)
Gauss's_inequality
Discrete probability distribution
Rademacher variables, including concentration inequalities such as Bernstein inequalities as well as anti-concentration inequalities like Tomaszewski's conjecture
Rademacher_distribution
Statistical inequality
}}\varepsilon ={\sqrt {\frac {\ln {\frac {2}{\alpha }}}{2n}}}.} Concentration inequality – a summary of bounds on sets of random variables. Dvoretzky, A
Dvoretzky–Kiefer–Wolfowitz inequality
Dvoretzky–Kiefer–Wolfowitz_inequality
Probabilistic inequality
{\frac {t}{np}}))} . Concentration inequality - a summary of tail-bounds on random variables. Bennett, G. (1962). "Probability Inequalities for the Sum of Independent
Bennett's_inequality
Causes of inequality may include executive compensation increasing relative to the average worker, financialization, greater industry concentration, lower
Income inequality in the United States
Income_inequality_in_the_United_States
Distribution of income or wealth between different groups
Economic inequality is an umbrella term for three concepts: income inequality, how the total sum of money paid to people is distributed among them; wealth
Economic_inequality
Theorem in probability theory
theory, Azuma's inequality or the Azuma–Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) gives a concentration result for the values
Azuma's_inequality
French statistician
Dvoretzky–Kiefer–Wolfowitz inequality, the Bousquet inequality, the concentration inequality, and the Efron-Stein inequality. With Lucien Birgé he worked
Pascal_Massart
Measure of inequality of a statistical distribution
wealth inequality in his work On the measurement of concentration and variability of characters in 1914. Here, he presented the concentration ratio, which
Gini_coefficient
Inequality in information theory
on hypothesis testing. (Bretagnolle–Huber–Carol Inequality is a variation of Concentration inequality for multinomially distributed random variables which
Bretagnolle–Huber_inequality
Uneven distribution of resources in a society
Social inequality occurs when resources within a society are distributed unevenly, often as a result of inequitable allocation practices that create distinct
Social_inequality
Theorem in probability theory
2211 works. Chernoff's inequality Edgeworth series List of inequalities List of mathematical theorems Concentration inequality Since the random variables
Berry–Esseen_theorem
ownership concentration was reported to be high even in areas where family farms are settled. This is an issue with regard to income inequality as family
Income_inequality_in_Brazil
Stochastic process
McDiarmid's inequality. The Doob martingale was introduced by Joseph L. Doob in 1940 to establish concentration inequalities such as McDiarmid's inequality, which
Doob_martingale
The inequality of wealth (i.e., inequality in the distribution of assets) has substantially increased in the United States since the late 1980s. Wealth
Wealth inequality in the United States
Wealth_inequality_in_the_United_States
Mathematical concept
space Fuzzy measure theory Filtered probability space Talagrand's concentration inequality Loève, Michel. Probability Theory, Vol 1. New York: D. Van Nostrand
Probability_space
Database of wealth and income distribution
World Inequality Database (WID), previously The World Wealth and Income Database, also known as WID.world, is an extensive, open and accessible database
World_Inequality_Database
2012 nonfiction book by Joseph Stiglitz
The Price of Inequality: How Today's Divided Society Endangers Our Future is a 2012 book by Joseph Stiglitz that deals with income inequality in the United
The_Price_of_Inequality
Mathematical result
2 ( k ) {\textstyle r\sim \chi ^{2}(k)} . Thus, it satisfies a concentration inequality for the chi-squared distribution: Pr ( r ∈ ( 1 ± ϵ ) k ) ≥ 1 −
Johnson–Lindenstrauss_lemma
Ownership of land in a particular area by a small number of people or organizations
a correlation between land concentration and deforestation. Another study found an inverse correlation between inequality of land ownership and economic
Concentration of land ownership
Concentration_of_land_ownership
Probabilistic inequality
and Adaptive Inference Based on Martingale Concentration (Thesis). Choi, K. P. (1988). "Some sharp inequalities for Martingale transforms". Transactions
Ville's_inequality
report observes that inequality in India has been on the rise since the early 2000s, particularly with a significant concentration of wealth among the
Income_inequality_in_India
Method used in statistics, pattern recognition, and other fields
measures on a multidimensional cube (this is proven using Talagrand's concentration inequality for product probability spaces). Data separability by classical
Linear_discriminant_analysis
Term in market economics
firms. Davies (1979) suggests that a concentration index should in general depend on both N and the inequality of firms' shares. Davis (1980) indicates
Market_concentration
Type of probability distribution
{\textstyle C_{a}>0} depends only on a {\textstyle a} . Gaussian concentration inequality for Lipschitz functions (Tao 2012, Theorem 2.1.12.)—If f : R n
Sub-Gaussian_distribution
Economic model showing fair trading leads to inequality
wealth concentration. Compared with other kinetic exchange models, the yard-sale model stands out for the extent to which it magnifies inequality. In models
Yard-sale_model
Technique to reduce dimensionality of points in Euclidean space
(k/2,(d-k)/2)} , with mean k / d {\displaystyle k/d} . We have a concentration inequality P r [ | ‖ v ‖ 2 − k d | ≥ ϵ k d ] ≤ 3 exp ( − k ϵ 2 / 64 ) {\displaystyle
Random_projection
Result in geometry
In mathematics, the Loomis–Whitney inequality is a result in geometry, which in its simplest form, allows one to estimate the "size" of a d {\displaystyle
Loomis–Whitney_inequality
Average value of a random variable
their joint density. Concentration inequalities control the likelihood of a random variable taking on large values. Markov's inequality is among the best-known
Expected_value
Theorem in geometry
mathematics, the Brunn–Minkowski theorem (or Brunn–Minkowski inequality) is an inequality relating the volumes (or more generally Lebesgue measures) of
Brunn–Minkowski_theorem
Indian mathematician (born 1979)
on the study of fluctuations in random structures, concentration and super-concentration inequalities, Poisson and other non-normal limits, first-passage
Sourav_Chatterjee
Ways inequality is measured
Income inequality metrics or income distribution metrics are used by social scientists to measure the distribution of income and economic inequality among
Income_inequality_metrics
Inclusion–exclusion principle Impossible event Information geometry Talagrand's concentration inequality Probability theory Probability space Sample space Standard probability
List_of_probability_topics
Branch of statistical computational learning theory
inequality relies on symmetrization, and then argues conditionally on the data using concentration inequalities (in particular Hoeffding's inequality)
Vapnik–Chervonenkis_theory
Idea and situation that women and men are not treated as equal
Gender inequality refers to gender specific disparities in access to opportunities, resources, rights, and protections. Where gender inequality is present
Gender_inequality
Inequality in probability theory
of variance proxy. Hoeffding's inequality Bennett's inequality Pascal Massart (26 April 2007). Concentration Inequalities and Model Selection: Ecole d'Eté
Hoeffding's_lemma
Existence theorem for economical additive bases of every order
Originally it was dealt with by means of Janson's inequality, a type of concentration inequality for multivariate polynomials. Tao & Vu (2006) present
Erdős–Tetali_theorem
Inequality between nations' wealth
International inequality refers to inequality between countries, as compared to global inequality, which is inequality between people across countries
International_inequality
Distribution of a country's total GDP amongst its population
government redistribution affects inequality. There is not a clear view on how long-run trends in income concentration are shaped by the major changes in
Income_distribution
Inequality across geographical regions
Spatial inequality refers to the unequal distribution of resources and income across geographical regions. Attributable to local differences in infrastructure
Spatial_inequality
The rich get richer and the poor get poorer
1968. Early studies of Matthew effects were primarily concerned with the inequality in the way scientists were recognized for their work. However, Norman
Matthew_effect
Imprisonment or confinement of groups of people without trial
prisons or in facilities known as internment camps or concentration camps. The term concentration camp originates from the Spanish–Cuban Ten Years' War
Internment
Problem in communication complexity theory
ISBN 9781450306911. S2CID 10274326. Vidick, Thomas (2012). "A concentration inequality for the overlap of a vector on a large set, with application to
Gap-Hamming_problem
2017 book by Noam Chomsky
single topic: income inequality. It lays out Chomsky's views and analysis of neoliberalism, and focuses on the rapid concentration of wealth and power
Requiem for the American Dream
Requiem_for_the_American_Dream
Science prizes established by Run Run Shaw
researchers in the field. 2019 Michel Talagrand France For his work on concentration inequalities, on suprema of stochastic processes and on rigorous results for
Shaw_Prize
Person who has at least one billion units of a currency
and seven in ten supported raising taxes on the richest. Oxfam's 2026 inequality report, released at the World Economic Forum in Davos, cited a World Values
Billionaire
French economist (born 1971)
Economics (PSE) and Centennial Professor of Economics in the International Inequalities Institute at the London School of Economics (LSE). Piketty's work focuses
Thomas_Piketty
Size of a mathematical ball
1080/0025570X.1989.11977419. Song Mei (2021-02-10). "Lecture 7: Concentration Inequalities and Field theoretic calculations" (PDF). www.stat.berkeley.edu
Volume_of_an_n-ball
Concept in sociology and economy
Malinen, Tuomas (2016-09-01). "Does income inequality contribute to credit cycles?". The Journal of Economic Inequality. 14 (3): 309–325. doi:10.1007/s10888-016-9334-6
Conspicuous_consumption
Measure of economic inequality
distribution or consumption inequality is an alternative to the income distribution or wealth distribution for judging economic inequality, comparing levels of
Consumption_distribution
Overview of and topical guide to statistics
Conditional probability Law of large numbers Central limit theorem Concentration inequality Convergence of random variables Computational statistics Markov
Outline_of_statistics
Political philosophy
focusing on within-country concentration of income and wealth. Branko Milanovic provided evidence of increasing inequality at the global level, showing
Redistribution of income and wealth
Redistribution_of_income_and_wealth
2013 book by economist Thomas Piketty
written by French economist Thomas Piketty. It focuses on wealth and income inequality in Europe and the United States since the 18th century. It was first published
Capital in the Twenty-First Century
Capital_in_the_Twenty-First_Century
and an inequality by Rudelson". arXiv:1004.3821 [math.PR]. Paulin, D.; Mackey, L.; Tropp, J. A. (2013). "Deriving Matrix Concentration Inequalities from
Matrix_Chernoff_bound
Personal Affluence wealth inequality Social class Income inequality gender pay gap racial pay gap Lists by income States (by inequality) Counties (highest /
Wealth_of_Elon_Musk
Effects of income inequality, researchers have found, include higher rates of health and social problems, and lower rates of social goods, a lower population-wide
Effects of economic inequality
Effects_of_economic_inequality
how wealth is distributed through asset transactions. AEMs illustrate inequality as an emergent property of systems of stochastically interacting agents
Asset_exchange_model
French statistician
statistician who has studied Hawkes processes, density estimation, and concentration inequalities, and applied them in neuroscience, neural connectivity reconstruction
Patricia_Reynaud-Bouret
American mathematician (born 1977)
has established a family of results, collectively called matrix concentration inequalities, that includes the matrix Chernoff bound. Tropp was a recipient
Joel_Tropp
Multi-armed bandit sequential game
far it is from optimal. Multi-armed bandit Design of experiments Concentration inequality Robertson, David S.; Lee, Kevin M.; López-Kolkovska, Beatriz C
Best_arm_identification
Department of University College London
Wendy Carlin James M. and Cathleen D. Stone Centre on Wealth Concentration, Inequality and the Economy – directed by Wendy Carlin and Imran Rasul The
UCL_Department_of_Economics
American computer scientist and statistician
systems, robust matrix factorization for video surveillance, and concentration inequalities for matrices. After Berkeley, he joined Stanford University, first
Lester_Mackey
1904–1908 genocide by the German Empire
extermination policy which led to the incarceration of all Herero people in concentration camps, where around half of those captured died due to disease, malnourishment
Herero_and_Nama_genocide
Inequality measure
(also known as the Atkinson measure or Atkinson inequality measure) is a measure of income inequality developed by British economist Anthony Barnes Atkinson
Atkinson_index
Political slogan
phrase directly refers to the income and wealth inequality in the United States, with a concentration of wealth among the top-earning 1%. It reflects
We_are_the_99%
Statistical technique developed by Alan Turing and I. J. Good
Learning Theory pp. 1–6 David A. McAllester, Ortiz, Luis (2003) Concentration Inequalities for the Missing Mass and for Histogram Rule Error, Journal of
Good–Turing frequency estimation
Good–Turing_frequency_estimation
Kuznets, the rise in inequality is unavoidable with the onset of the Industrial Revolution, as it requires a dense concentration of capital to enable
History of economic inequality
History_of_economic_inequality
Abundance of financial assets or possessions
human capital to be $1,152 trillion. According to the Kuznets curve, inequality of wealth and income increases during the early phases of economic development
Wealth
Gender inequality in India refers to health, education, economic and political inequalities between men and women in India. Various international gender
Gender_inequality_in_India
Graph of wealth or income distribution
measures of concentration for any dataset Free Calculator: Online and downloadable scripts (Python and Lua) for Atkinson, Gini, and Hoover inequalities Users
Lorenz_curve
Society controlled by business corporations
also increase income inequality. Nobel Prize winner of economics Joseph Stiglitz wrote in May 2011: "Much of today’s inequality is due to manipulation
Corporatocracy
Aphorism due to Percy Bysshe Shelley
wording, as a summary of the effect of free market capitalism on excessive inequality. Andrew Jackson, the seventh President of the U.S. (1829–1837), in his
The rich get richer and the poor get poorer
The_rich_get_richer_and_the_poor_get_poorer
The Gaussian isoperimetric inequality also follows from Ehrhard's inequality. Concentration of measure Borell–TIS inequality Sudakov, V. N.; Tsirel'son
Gaussian isoperimetric inequality
Gaussian_isoperimetric_inequality
British economist (1944–2017)
single-handedly established the modern British field of inequality and poverty studies. He worked on inequality and poverty for over four decades. Atkinson was
Tony_Atkinson
J., Jarman, J., and Blackburn, R. M. Gender Inequality in the Labor Market: Occupational Concentration and Segregation. 1995. International Labour Organization
Gender_inequality_in_Thailand
1996 book
Inequality by Design: Cracking the Bell Curve Myth is a 1996 book by Claude S. Fischer, Michael Hout, Martín Sánchez Jankowski, Samuel R. Lucas, Ann Swidler
Inequality_by_Design
Imperial German concentration camp in Luderitz, colonial Namibia (1905–1907)
-26.64583; 15.15389 Shark Island or "Death Island" was one of five concentration camps in German South West Africa. It was located on Shark Island off
Shark Island concentration camp
Shark_Island_concentration_camp
Relates income inequality and income mobility
cross-sectional income inequality and persistence of income across generations. The scatter plot shows a correlation between income inequality in a country and
Great_Gatsby_Curve
Probability distribution
non-negative and sum up to 1. The concentration weight in this case is larger by a factor of K than the concentration parameter for a symmetric Dirichlet
Dirichlet_distribution
Historical period in the Southern United States from 1815 to 1861
manufacturing base lagged behind that of the non-slave states. Wealth inequality grew as the larger landholders took the greater share of the profits generated
Antebellum_South
Dominant post-WWII post-industrial global economy
manufacturing units supplying specialized markets. Increasing economic inequality of income, wealth and consumption. Consumerism on credit and the increasing
Late_capitalism
Inequality of access to information and communication technologies
motivational, material, skills, and usage access. The digital divide worsens inequality in access to information and resources. According to 2026 data from the
Digital_divide
CONCENTRATION INEQUALITY
CONCENTRATION INEQUALITY
Girl/Female
Tamil
Concentration, Ecstasy
Boy/Male
Hindu, Indian, Punjabi
Concentration
Boy/Male
Tamil
Thanmayee | தாநà¯à®®à®¾à®¯à¯€
Concentration, Ecstasy
Thanmayee | தாநà¯à®®à®¾à®¯à¯€
Boy/Male
Hindu, Indian
Concentration
Boy/Male
Hindu, Indian
Concentration
Boy/Male
Hindu, Indian, Sanskrit
Concentration; Meditation
Girl/Female
Tamil
Dheeksha | திகà¯à®·à®¾Â
Teaching, Concentration
Dheeksha | திகà¯à®·à®¾Â
Girl/Female
Indian
Teaching, Concentration
Boy/Male
Tamil
Concentration, Ecstasy
Girl/Female
Hindu, Indian, Sanskrit
Concentration
Boy/Male
Muslim
Imagination, Concentration
Girl/Female
Gujarati, Hindu, Indian, Kannada, Kashmiri, Malayalam, Modern, Traditional
Faith; Concentration
Boy/Male
Hindu, Indian, Kannada
Concentration
Boy/Male
Hindu
Concentration, Ecstasy
Boy/Male
Tamil
Concentration
Girl/Female
Indian, Telugu
Good Concentration
Girl/Female
Hindu
Concentration, Ecstasy
Boy/Male
Indian
Concentration
Girl/Female
Gujarati, Indian, Telugu
Trust; Concentration
Boy/Male
Hindu
Concentration, Ecstasy
CONCENTRATION INEQUALITY
CONCENTRATION INEQUALITY
Female
Hindi/Indian
(सà¥à¤¦à¤°à¥à¤¶à¤¨à¤¾) Feminine form of Hindi Sudarshan, SUDARSHANA means "seeing one's self correctly; right vision."
Boy/Male
Tamil
Travelling
Girl/Female
Hindu
Unity with friendship
Male
Dutch
, victor of the people.
Male
Spanish
Spanish and Portuguese form of Old High German Hrodric, RODRIGO means "famous power."
Girl/Female
American, Australian
A Combination of the Prefix B and Riley
Female
English
English name borrowed from the name of an Italian island where Napoleon was exiled, derived from Latin Ilva, from Greek Aethale, ELBA means "soot, grime."
Boy/Male
Tamil
Powerful
Boy/Male
Arabic, Muslim
Gold
Boy/Male
Tamil
Charmin | சாரà¯à®®à¯€à®¨
Sport
CONCENTRATION INEQUALITY
CONCENTRATION INEQUALITY
CONCENTRATION INEQUALITY
CONCENTRATION INEQUALITY
CONCENTRATION INEQUALITY
n.
Strife; contention.
a.
Assembling; collecting; used for gathering or concentrating.
n.
Act of deconcentrating.
n.
A central point; a point of concentration.
a.
Serving or tending to concentrate; characterized by concentration.
p. pr. & vb. n.
of Concentrate
n.
The act or practice of collecting or concentrating the mind; concentration; self-control.
n.
The act or process of concentrating; the process of becoming concentrated, or the state of being concentrated; concentration.
n.
The faculty or propensity which has to do with concentrating the intellectual the intellectual powers.
n.
An apparatus for the separation of dry comminuted ore, by exposing it to intermittent puffs of air.
n.
A giddy or thoughtless person; one incapable of concentration or attention.
n.
Concentration of one's interests on one's self; self-love; selfishness.
v. t.
To withdraw from concentration; to decentralize.
n.
The quality of concentrating.
n.
The act or process of reducing the volume of a liquid, as by evaporation.
n.
A machine for concentrating ore. See Frue vanner.
n.
Content; satisfaction.
v. t.
To bring to a certain strength or grade of concentration; as, to gradate a saline solution.
n.
The act or process of removing the dress of ore and of reducing the valuable part to smaller compass, as by currents of air or water.
n.
A conjunction or concentration of rays in one point.