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Branch of probability theory
In probability theory, the theory of large deviations concerns the asymptotic behaviour of remote tails of sequences of probability distributions. While
Large_deviations_theory
Theorem
mathematics — specifically, in large deviations theory — the contraction principle is a theorem that states how a large deviation principle on one space "pushes
Contraction principle (large deviations theory)
Contraction_principle_(large_deviations_theory)
Fundamental result in the theory of large deviations
Cramér's theorem is a fundamental result in the theory of large deviations, a subdiscipline of probability theory. It determines the rate function of a series
Cramér's theorem (large deviations)
Cramér's_theorem_(large_deviations)
Theorem in mathematics
In mathematics, Laplace's principle is a basic theorem in large deviations theory which is similar to Varadhan's lemma. It gives an asymptotic expression
Laplace principle (large deviations theory)
Laplace_principle_(large_deviations_theory)
Probability function
large deviations theory, a rate function is a function used to quantify the probabilities of rare events. Such functions are used to formulate large deviation
Rate_function
In mathematics, Varadhan's lemma is a result from the large deviations theory named after S. R. Srinivasa Varadhan. The result gives information on the
Varadhan's_lemma
Concept in stochastic analysis
the Contraction principle in large deviations theory reduces Freidlin–Wentzell's problem to demonstrating the large deviation principle for ( t , ε B t )
Rough_path
Mathematical formula
specifically, in large deviations theory — the tilted large deviation principle is a result that allows one to generate a new large deviation principle from
Tilted large deviation principle
Tilted_large_deviation_principle
Theorem in the large deviations theory of stochastic processes
to Mark Freidlin and Alexander D. Wentzell) is a result in the large deviations theory of stochastic processes. Roughly speaking, the Freidlin–Wentzell
Freidlin–Wentzell_theorem
divergence Le Cam's theorem Large deviations theory Contraction principle (large deviations theory) Varadhan's lemma Tilted large deviation principle Rate function
List_of_probability_topics
Generalization of the binomial distribution
In probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts
Multinomial_distribution
^{n}} to functional Wiener integration. The theorem is used in the large deviations theory of stochastic processes. Roughly speaking, out of Schilder's theorem
Schilder's_theorem
Mathematical theorem
distribution. In the language of large deviations theory, Sanov's theorem identifies the rate function for large deviations of the empirical measure of a
Sanov's_theorem
standard deviations. This is a large deviation. Though rare in a small domain (of space or/and time), large deviations may be quite usual in a large domain
Large deviations of Gaussian random functions
Large_deviations_of_Gaussian_random_functions
Mathematical result in large deviations theory
is a result in large deviations theory. Heuristically speaking, the Dawson–Gärtner theorem allows one to transport a large deviation principle on a “smaller”
Dawson–Gärtner_theorem
tests are computed using Sanov's theorem and other results from large deviations theory. There are various methods used to show that an error exponent
Error exponents in hypothesis testing
Error_exponents_in_hypothesis_testing
Russian-American probability theorist
Freidlin–Wentzell theory, which is an important part of the large deviations theory. Freidlin and Wentzell are the authors of the first monograph on the large deviations
Mark_Freidlin
Study of convergence properties of statistical estimators
In statistics, asymptotic theory, or large sample theory, is a framework for assessing properties of estimators and statistical tests. Within this framework
Asymptotic theory (statistics)
Asymptotic_theory_(statistics)
Topic in mathematics
number of samples. Such results are studied in large deviations theory; intuitively, it is the large deviations that would violate equipartition, but these
Asymptotic equipartition property
Asymptotic_equipartition_property
Welsh mathematical physicist, worked in Ireland
including quantum measurement, Bose–Einstein condensation and large deviations theory. He was a senior professor at the Dublin Institute for Advanced
John_T._Lewis
Measure of variation in statistics
within one standard deviation of the average, 95% within two standard deviations, and 99.7% within three. The standard deviation of a random variable
Standard_deviation
Mathematical concept
{\displaystyle a} and b . {\displaystyle b.} This estimate is useful in large deviations theory under exponential moment conditions, because b ln b {\displaystyle
Young's inequality for products
Young's_inequality_for_products
Representation of a type of random process
has been suppressed by assuming that the variable has been measured as deviations from its mean) as X t = 1 φ ( B ) ε t . {\displaystyle X_{t}={\frac {1}{\varphi
Autoregressive_model
Equivalence relation on mathematical measures
probability measures are "the same" from the point of view of large deviations theory. Let ( M , d ) {\displaystyle (M,d)} be a metric space and consider
Exponentially equivalent measures
Exponentially_equivalent_measures
L-theory the K-theory of quadratic forms. Large deviations theory part of probability theory studying events of small probability (tail events). Large sample
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Branch of statistics focusing on large deviations
Fisher–Tippett–Gnedenko theorem Generalized extreme value distribution Large deviation theory Outlier Pareto distribution Pickands–Balkema–de Haan theorem Rare
Extreme_value_theory
Method for approximate evaluation of integrals
descent Saddlepoint approximation method Large deviations theory Laplace principle (large deviations theory) Laplace's approximation Tierney, Luke; Kadane
Laplace's_method
Notion for convergence of metric spaces
the concept of Gromov–Hausdorff limits is closely related to large-deviations theory. Intrinsic flat distance David A. Edwards, "The Structure of Superspace"
Gromov–Hausdorff_convergence
Mathematical transformation
for instance, the same "pull" between a capacitor's plates. In large deviations theory, the rate function is defined as the Legendre transformation of
Legendre_transformation
Topics referred to by the same term
a German xDT format to transfer laboratory tests Large deviations theory, field of probability theory Learning Design and Technology, an academic program
LDT
Stochastic volatility model used in derivatives markets
implement in computer code, and lends itself well to risk management of large portfolios of options in real time. It is convenient to express the solution
SABR_volatility_model
Laplace's rule of succession Laplace smoothing Laplace principle (large deviations theory) Laplace series Laplace transform Two-sided Laplace transform Laplace–Carson
List of things named after Pierre-Simon Laplace
List_of_things_named_after_Pierre-Simon_Laplace
Mathematics of convex functions and sets
probability measures by testing them against convex functions, and large deviations theory uses the Legendre–Fenchel transform to express rate functions as
Convex_analysis
2009 book on combinatorial enumeration
combinatorial quantities of interest, it also studies limit theorems and large deviations theory for these quantities. Three appendices provide background on combinatorics
Analytic_Combinatorics_(book)
American mathematician
1955) is an American mathematician, a pioneer in the usage of large deviations theory in performance evaluation and related areas. Weiss received his
Alan_Weiss_(mathematician)
Israeli mathematician
filtering theory with applications to control theory (electrical engineering), the spectral theory of random matrices, the theory of large deviations in probability
Ofer_Zeitouni
Averages of repeated trials converge to the expected value
In probability theory, the law of large numbers is a mathematical law which states that the average of the results obtained from a large number of independent
Law_of_large_numbers
Dutch mathematician (born 1970)
contributed to the book Queues and Lévy fluctuation theory, published in 2015. "Large deviations for Gaussian queues" (2007) Michel Mandjes at the Mathematics
Michel_Mandjes
Russian mathematician (1931–2026)
theory, mathematical statistics, stochastic processes, queueing theory, large deviations, random walks, and asymptotic methods. He authored several influential
Aleksandr_Borovkov
Solution to a stochastic differential equation
In probability theory and statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Diffusion
Diffusion_process
Statistical measure
Mean absolute error Average absolute deviation Mean signed deviation Mean squared deviation Squared deviations Errors and residuals in statistics Coefficient
Root_mean_square_deviation
On eigenvalues of random matrices
In probability theory, more specifically the study of random matrices, the circular law concerns the distribution of eigenvalues of an n × n {\displaystyle
Circular_law
distribution Laplace principle (large deviations theory) LaplacesDemon – software Large deviations theory Large deviations of Gaussian random functions LARS
List_of_statistics_articles
Irish educational institution and academic publisher
Lewis had interests including Bose–Einstein condensation and Large deviations theory.[citation needed] The school has three senior professors at present:[when
Dublin Institute for Advanced Studies
Dublin_Institute_for_Advanced_Studies
Indian American mathematician (born 1940)
probability theory and in particular for creating a unified theory of large deviations. He is regarded as one of the fundamental contributors to the theory of
S._R._Srinivasa_Varadhan
Statistical error measure
squared error, the equivalent for mean absolute error is least absolute deviations. MAE is not identical to root-mean square error (RMSE), although some
Mean_absolute_error
Statistical measure of variability
The absolute deviations about 2 are (1, 1, 0, 0, 2, 4, 7) which in turn have a median value of 1 (because the sorted absolute deviations are (0, 0, 1
Median_absolute_deviation
Concept in statistics
way of constructing a GRF is by assuming that the field is the sum of a large number of plane, cylindrical or spherical waves with uniformly distributed
Gaussian_random_field
Mathematical approach to quantum physics
expansion parameter) becomes too large, violating the requirement that corrections must be small. Perturbation theory also fails to describe states that
Perturbation theory (quantum mechanics)
Perturbation_theory_(quantum_mechanics)
Macroeconomic model
that the deviations in real GNP are comparatively small and might be attributable to measurement errors rather than real deviations. We call large positive
Real_business-cycle_theory
American mathematician
Massachusetts Amherst in 1975. In 1984, he improved a key result in large deviations theory originally due to Jürgen Gärtner, which is now known as Gärtner-Ellis
Richard_S._Ellis
anl Large deviations theory Contraction principle Cramér's theorem Exponentially equivalent measures Freidlin–Wentzell theorem Laplace principle Large deviations
Catalog of articles in probability theory
Catalog_of_articles_in_probability_theory
Concept in measure theory
the concept of exponential tightness, which has applications in large deviations theory. A family of probability measures ( μ δ ) δ > 0 {\displaystyle
Tightness_of_measures
Brazilian mathematical statistician and probability theorist
theorist who is known for her expertise in stochastic processes and large deviations theory. She is a professor of statistics in the Institute of Mathematics
Maria_Eulália_Vares
Theory of behavioral economics
theory emerged from earlier research by Kahneman and Tversky in the 1970s on heuristics and cognitive biases, which documented systematic deviations from
Prospect_theory
Swedish mathematician (1893–1985)
statistics and probabilistic number theory. John Kingman described him as "one of the giants of statistical theory". Harald Cramér was born in Stockholm
Harald_Cramér
Mathematicians of Welsh nationality
Contributed to quantum measurement, Bose–Einstein condensation and large deviations theory. William Morgan 26 May 1750 Bridgend 4 May 1833 London Considered
List_of_Welsh_mathematicians
Israeli computer scientist
information theory. He applied Compression Algorithms based on Approximate String Matching. He presented performance analysis based on large deviations theory (LDT)
Ilan_Sadeh
information theory and statistics, Kullback's inequality is a lower bound on the Kullback–Leibler divergence expressed in terms of the large deviations rate
Kullback's_inequality
Topics referred to by the same term
principle may refer to: Contraction principle (large deviations theory), a theorem that states how a large deviation principle on one space "pushes forward"
Contraction_principle
Bound on probability of a random variable being far from its mean
75% of values must lie within two standard deviations of the mean and 88.88% within three standard deviations for a broad range of different probability
Chebyshev's_inequality
Relative measure of dispersion expressed as the ratio of standard deviation to the mean
In probability theory and statistics, the coefficient of variation (CV), also known as normalized root-mean-square deviation (NRMSD), and relative standard
Coefficient_of_variation
Topics referred to by the same term
distributed random variable Cramér's theorem (large deviations), a fundamental result in the theory of large deviations Cramer's theorem (algebraic curves), a
Cramér's_theorem
Israeli-American mathematician
research deals with probability theory and stochastic processes, the theory of large deviations, the spectral theory of random matrices, random walks
Amir_Dembo
Falsifiable explanation of natural phenomena
A scientific theory is an explanation of an aspect of the natural world that can be or that has been repeatedly tested and has corroborating evidence in
Scientific_theory
Methods of mathematical approximation
In mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related
Perturbation_theory
In probability theory and statistics, a continuous-time stochastic process, or a continuous-space-time stochastic process is a stochastic process for which
Continuous-time stochastic process
Continuous-time_stochastic_process
Theory of response to surprise events
that tell you close to nothing. Why? Because the bell curve ignores large deviations, cannot handle them, yet makes us confident that we have tamed uncertainty
Black_swan_theory
Mathematical framework for investment risk
correlated — then the portfolio return's standard deviation is the sum of the asset returns' standard deviations weighted by the fractions held in the portfolio
Modern_portfolio_theory
Law of thermodynamics for vapour pressure of a mixture
theoretically possible, as actual examples of mixed deviation exist. The possible physical deviations are not entirely arbitrary however, as they are constrained
Raoult's_law
Unexpectedly large transient ocean surface wave
early evidence that waves could grow significantly larger than anticipated by conventional theories of wave breaking. This work highlighted that in cases
Rogue_wave
black-body to very high precision; deviations do not exceed 2 parts in 100000. This showed that earlier claims of spectral deviations were incorrect, and essentially
History of the Big Bang theory
History_of_the_Big_Bang_theory
Quasiparticle in condensed matter physics
Srinivasa Varadhan, applying large deviation theory to the path integral formulation for the self-energy, showed the large α exactitude of this Landau–Pekar
Polaron
Mathematical models of strategic interactions
Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively
Game_theory
Description of physical properties at the atomic and subatomic scale
for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion
Quantum_mechanics
Theory in evolutionary biology proposed to explain sex ratio deviations in mammals
pressures exist to maintain a 1:1 sex ratio, evolution will favor local deviations from this if one sex has a likely greater reproductive payoff than is
Trivers–Willard_hypothesis
Society, Abel Prize winner. Pioneer of LargeDeviations Theory. C. P. Ramanujam (1938–1974), worked on number theory and algebraic geometry T. S. Vijayaraghavan
List_of_Tamil_people
American television sitcom (2007–2019)
The Big Bang Theory is an American television sitcom created by Chuck Lorre and Bill Prady for CBS. It aired from September 24, 2007, to May 16, 2019,
The_Big_Bang_Theory
Procedure to estimate standard deviation from a sample
statistical theory, unbiased estimation of a standard deviation is the calculation from a statistical sample of an estimated value of the standard deviation (a
Unbiased estimation of standard deviation
Unbiased_estimation_of_standard_deviation
Physical theory of the cosmos
one of the underlying principles of the theory of relativity. The cosmological principle states that on large scales the universe is homogeneous and isotropic—appearing
Big_Bang
American mathematician (1940–2025)
Springer. 1979.; reprintings 1997, 2006 An introduction to the theory of large deviations. Springer-Verlag. 1984. with Andrzej Korzeniowski: Korzeniowski
Daniel_W._Stroock
Branch of multiobjective optimization
target value to be achieved. Deviations are measured from these goals both above and below the target. Unwanted deviations from this set of target values
Goal_programming
Stochastic process in probability theory
In probability theory, an empirical process is a stochastic process that characterizes the deviation of the empirical distribution function from its expectation
Empirical_process
Function in fluid mathematics
indicates the deviation from statically neutral state, with smaller | L | {\displaystyle |L|} values corresponding to larger deviations from neutral conditions
Monin–Obukhov similarity theory
Monin–Obukhov_similarity_theory
French mathematician (born 1969)
Arous & A. Guionnet (1997). "Large deviations for Wigner's law and Voiculescu's non-commutative entropy". Probab. Theory Relat. 108 (4): 517–542. doi:10
Alice_Guionnet
The Big Bang Theory is an American television sitcom created and executively produced by Chuck Lorre and Bill Prady for CBS. Like the name of the series
List of The Big Bang Theory episodes
List_of_The_Big_Bang_Theory_episodes
Risk of statistically extreme events
three standard deviations may also be broadened, such as the SKEW index which uses the larger tail region starting at two standard deviations. Although tail
Tail_risk
Numbers significantly larger than those used regularly
({n+1})^{k_{n+1}}} is rewritten. For describing numbers approximately, deviations from the decreasing order of values of n are not needed. For example,
Large_numbers
Classical statement of gravity as force
sufficiently accurate for many practical purposes and is therefore widely used. Deviations from it are small when the dimensionless quantities ϕ / c 2 {\displaystyle
Newton's law of universal gravitation
Newton's_law_of_universal_gravitation
American theoretical physicist and cosmologist
Bang theory found values of omega near 1 to be puzzling, because any deviations from 1 would quickly become much, much larger. In inflation theory, no
Alan_Guth
Observation far apart from others in statistics and data science
standard deviations from the mean, p is approximately 0.3%, and thus for 1000 trials one can approximate the number of samples whose deviation exceeds
Outlier
Conspiracy theories about the 1983 shootdown
Korean Air Lines Flight 007 alternative theories concerns the various theories put forward regarding the shooting down of Korean Air Lines Flight 007.
Korean Air Lines Flight 007 alternative theories
Korean_Air_Lines_Flight_007_alternative_theories
Swiss mathematician (born 1945)
martingale convergence theorems, combinatorial limit theorems, and the large deviations theory. Later in his career he dealt with stochastic models in mathematical
Erwin_Bolthausen
Theory of quantum gravity merging quantum mechanics and general relativity
loop quantum gravity)? Are there deviations from the predictions of general relativity at very small or very large scales or in other extreme circumstances
Loop_quantum_gravity
Theory of rapid universe expansion
"for pioneering the theory of cosmic inflation". It was developed further in the early 1980s. It explains the origin of the large-scale structure of the
Cosmic_inflation
Physicist
Lobo, J., Strumsky, D., & West, G. B. (2010). Urban scaling and its deviations: Revealing the structure of wealth, innovation and crime across cities
Luís_M._A._Bettencourt
Business process improvement technique
process standard deviation goes up, or the mean of the process moves away from the center of the tolerance, fewer standard deviations will fit between
Six_Sigma
Method of examining human decision-making
experimental psychology. Experiments include testing deviations from typical simplifications of economic theory such as the independence axiom and neglect of
Behavioral_game_theory
Attraction of masses and energy
redshift or the deviation of light by matter and gives values for the precession of Mercury which are incorrect. A vector field theory predicts negative
Gravity
Larson, David Royal (eds.). Large Deviations for Additive Functionals of Markov Chains: The 25th Great Plains Operator Theory Symposium, June 7–12, 2005
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
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