Markov–Krein theorem
In probability theory, the Markov–Krein theorem gives the best upper and lower bounds on the expected values of certain functions of a random variable where only the first moments of the random variab
Bayes' theorem
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions
Kakutani's theorem (measure theory)
In measure theory, a branch of mathematics, Kakutani's theorem is a fundamental result on the equivalence or mutual singularity of countable product measures. It gives an "if and only if" characterisa
Helly–Bray theorem
In probability theory, the Helly–Bray theorem relates the weak convergence of cumulative distribution functions to the convergence of expectations of certain measurable functions. It is named after Ed
Tilted large deviation principle
In mathematics — specifically, in large deviations theory — the tilted large deviation principle is a result that allows one to generate a new large deviation principle from an old one by "tilting", i
Law of large numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results
Cramér–Wold theorem
In mathematics, the Cramér–Wold theorem in measure theory states that a Borel probability measure on is uniquely determined by the totality of its one-dimensional projections. It is used as a method f
Dawson–Gärtner theorem
In mathematics, the Dawson–Gärtner theorem is a result in large deviations theory. Heuristically speaking, the Dawson–Gärtner theorem allows one to transport a large deviation principle on a “smaller”
Dynkin's formula
In mathematics — specifically, in stochastic analysis — Dynkin's formula is a theorem giving the expected value of any suitably smooth statistic of an Itō diffusion at a stopping time. It may be seen
Burke's theorem
In queueing theory, a discipline within the mathematical theory of probability, Burke's theorem (sometimes the Burke's output theorem) is a theorem (stated and demonstrated by while working at Bell Te
Leftover hash lemma
The leftover hash lemma is a lemma in cryptography first stated by Russell Impagliazzo, Leonid Levin, and Michael Luby. Imagine that you have a secret key X that has n uniform random bits, and you wou
Varadhan's lemma
In mathematics, Varadhan's lemma is a result from large deviations theory named after S. R. Srinivasa Varadhan. The result gives information on the asymptotic distribution of a statistic φ(Zε) of a fa
Bapat–Beg theorem
In probability theory, the Bapat–Beg theorem gives the joint probability distribution of order statistics of independent but not necessarily identically distributed random variables in terms of the cu
Lukacs's proportion-sum independence theorem
In statistics, Lukacs's proportion-sum independence theorem is a result that is used when studying proportions, in particular the Dirichlet distribution. It is named after Eugene Lukacs.
Palm–Khintchine theorem
In probability theory, the Palm–Khintchine theorem, the work of Conny Palm and Aleksandr Khinchin, expresses that a large number of renewal processes, not necessarily Poissonian, when combined ("super
Infinite monkey theorem
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, such as the complete works of Wi
Markov chain central limit theorem
In the mathematical theory of random processes, the Markov chain central limit theorem has a conclusion somewhat similar in form to that of the classic central limit theorem (CLT) of probability theor
Skorokhod's embedding theorem
In mathematics and probability theory, Skorokhod's embedding theorem is either or both of two theorems that allow one to regard any suitable collection of random variables as a Wiener process (Brownia
Doob–Dynkin lemma
In probability theory, the Doob–Dynkin lemma, named after Joseph L. Doob and Eugene Dynkin, characterizes the situation when one random variable is a function of another by the inclusion of the -algeb
Fernique's theorem
In mathematics - specifically, in measure theory - Fernique's theorem is a result about Gaussian measures on Banach spaces. It extends the finite-dimensional result that a Gaussian random variable has
Glivenko's theorem (probability theory)
In probability theory, Glivenko's theorem states that if , are the characteristic functions of some probability distributions respectively and almost everywhere, then in the sense of probability distr
Heyde theorem
In the mathematical theory of probability, the Heyde theorem is the characterization theorem concerning the normal distribution (the Gaussian distribution) by the symmetry of one linear form given ano
Wiener–Khinchin theorem
In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation
Bartlett's theorem
In queueing theory, Bartlett's theorem gives the distribution of the number of customers in a given part of a system at a fixed time.
Continuous mapping theorem
In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random variables. A continuous function, in Heine’s defi
Lévy's modulus of continuity theorem
Lévy's modulus of continuity theorem is a theorem that gives a result about an almost sure behaviour of an estimate of the modulus of continuity for Wiener process, that is used to model what's known
Jury theorem
A jury theorem is a mathematical theorem proving that, under certain assumptions, a decision attained using majority voting in a large group is more likely to be correct than a decision attained by a
Slutsky's theorem
In probability theory, Slutsky’s theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables. The theorem was named after Eugen Slu
Law of total probability
In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome wh
Maxwell's theorem
In probability theory, Maxwell's theorem, named in honor of James Clerk Maxwell, states that if the probability distribution of a vector-valued random variable X = ( X1, ..., Xn )T is the same as the
Uniform convergence in probability
Uniform convergence in probability is a form of convergence in probability in statistical asymptotic theory and probability theory. It means that, under certain conditions, the empirical frequencies o
Coupon collector's problem
In probability theory, the coupon collector's problem describes "collect all coupons and win" contests. It asks the following question: If each box of a brand of cereals contains a coupon, and there a
Poisson limit theorem
In probability theory, the law of rare events or Poisson limit theorem states that the Poisson distribution may be used as an approximation to the binomial distribution, under certain conditions. The
Kolmogorov's zero–one law
In probability theory, Kolmogorov's zero–one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, namely a tail event of independent σ-algebras, will either al
Maximal ergodic theorem
The maximal ergodic theorem is a theorem in ergodic theory, a discipline within mathematics. Suppose that is a probability space, that is a (possibly noninvertible) measure-preserving transformation,
Contraction principle (large deviations theory)
In mathematics — specifically, in large deviations theory — the contraction principle is a theorem that states how a large deviation principle on one space "pushes forward" (via the pushforward of a p
Raikov's theorem
Raikov’s theorem, named for Russian mathematician Dmitrii Abramovich Raikov, is a result in probability theory. It is well known that if each of two independent random variables ξ1 and ξ2 has a Poisso
De Finetti's theorem
In probability theory, de Finetti's theorem states that exchangeable observations are conditionally independent relative to some latent variable. An epistemic probability distribution could then be as
Cox's theorem
Cox's theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates. This derivation justifies the so-called "logical" in
Gleason's theorem
In mathematical physics, Gleason's theorem shows that the rule one uses to calculate probabilities in quantum physics, the Born rule, can be derived from the usual mathematical representation of measu
Condorcet's jury theorem
Condorcet's jury theorem is a political science theorem about the relative probability of a given group of individuals arriving at a correct decision. The theorem was first expressed by the Marquis de
Tanaka's formula
In the stochastic calculus, Tanaka's formula states that where Bt is the standard Brownian motion, sgn denotes the sign function and Lt is its local time at 0 (the local time spent by B at 0 before ti
Wendel's theorem
In geometric probability theory, Wendel's theorem, named after James G. Wendel, gives the probability that N points distributed uniformly at random on an -dimensional hypersphere all lie on the same "
Cramér's theorem (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 of iid random variables.A weak version of
Kolmogorov's two-series theorem
In probability theory, Kolmogorov's two-series theorem is a result about the convergence of random series. It follows from Kolmogorov's inequality and is used in one proof of the strong law of large n
Structure theorem for Gaussian measures
In mathematics, the structure theorem for Gaussian measures shows that the abstract Wiener space construction is essentially the only way to obtain a strictly positive Gaussian measure on a separable
Algorithmic Lovász local lemma
In theoretical computer science, the algorithmic Lovász local lemma gives an algorithmic way of constructing objects that obey a system of constraints with limited dependence. Given a finite set of ba
Laplace principle (large deviations theory)
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 for the Lebesgue integral of exp(−θφ(x)) over
Martingale representation theorem
In probability theory, the martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of a
Engelbert–Schmidt zero–one law
The Engelbert–Schmidt zero–one law is a theorem that gives a mathematical criterion for an event associated with a continuous, non-decreasing additive functional of Brownian motion to have probability
Schuette–Nesbitt formula
In mathematics, the Schuette–Nesbitt formula is a generalization of the inclusion–exclusion principle. It is named after and Cecil J. Nesbitt. The probabilistic version of the Schuette–Nesbitt formula
Craps principle
In probability theory, the craps principle is a theorem about event probabilities under repeated iid trials. Let and denote two mutually exclusive events which might occur on a given trial. Then the p
Kolmogorov's three-series theorem
In probability theory, Kolmogorov's Three-Series Theorem, named after Andrey Kolmogorov, gives a criterion for the almost sure convergence of an infinite series of random variables in terms of the con
Spitzer's formula
In probability theory, Spitzer's formula or Spitzer's identity gives the joint distribution of partial sums and maximal partial sums of a collection of random variables. The result was first published
Le Cam's theorem
In probability theory, Le Cam's theorem, named after Lucien Le Cam (1924 – 2000), states the following. Suppose:
* are independent random variables, each with a Bernoulli distribution (i.e., equal to
Bertrand's ballot theorem
In combinatorics, Bertrand's ballot problem is the question: "In an election where candidate A receives p votes and candidate B receives q votes with p > q, what is the probability that A will be stri
Pickands–Balkema–De Haan theorem
The Pickands–Balkema–De Haan theorem gives the asymptotic tail distribution of a random variable, when its true distribution is unknown. It is often called the second theorem in extreme value theory.
Cramér's decomposition theorem
Cramér’s decomposition theorem for a normal distribution is a result of probability theory. It is well known that, given independent normally distributed random variables ξ1, ξ2, their sum is normally
Jessen–Wintner theorem
In mathematics, the Jessen–Wintner theorem, introduced by Jessen and Wintner, asserts that a random variable of Jessen–Wintner type, meaning the sum of an almost surely convergent series of independen
Aumann's agreement theorem
Aumann's agreement theorem was stated and proved by Robert Aumann in a paper titled "Agreeing to Disagree", which introduced the set theoretic description of common knowledge. The theorem concerns age
Hammersley–Clifford theorem
The Hammersley–Clifford theorem is a result in probability theory, mathematical statistics and statistical mechanics that gives necessary and sufficient conditions under which a strictly positive prob
Feldman–Hájek theorem
In probability theory, the Feldman–Hájek theorem or Feldman–Hájek dichotomy is a fundamental result in the theory of Gaussian measures. It states that two Gaussian measures and on a locally convex spa
Borel–Cantelli lemma
In probability theory, the Borel–Cantelli lemma is a theorem about sequences of events. In general, it is a result in measure theory. It is named after Émile Borel and Francesco Paolo Cantelli, who ga
Disintegration theorem
In mathematics, the disintegration theorem is a result in measure theory and probability theory. It rigorously defines the idea of a non-trivial "restriction" of a measure to a measure zero subset of
Kosambi–Karhunen–Loève theorem
In the theory of stochastic processes, the Karhunen–Loève theorem (named after Kari Karhunen and Michel Loève), also known as the Kosambi–Karhunen–Loève theorem is a representation of a stochastic pro
Doob–Meyer decomposition theorem
The Doob–Meyer decomposition theorem is a theorem in stochastic calculus stating the conditions under which a submartingale may be decomposed in a unique way as the sum of a martingale and an increasi
Anderson's theorem
In mathematics, Anderson's theorem is a result in real analysis and geometry which says that the integral of an integrable, symmetric, unimodal, non-negative function f over an n-dimensional convex bo
Cameron–Martin theorem
In mathematics, the Cameron–Martin theorem or Cameron–Martin formula (named after Robert Horton Cameron and W. T. Martin) is a theorem of measure theory that describes how abstract Wiener measure chan
Reflection principle (Wiener process)
In the theory of probability for stochastic processes, the reflection principle for a Wiener process states that if the path of a Wiener process f(t) reaches a value f(s) = a at time t = s, then the s
Binomial sum variance inequality
The binomial sum variance inequality states that the variance of the sum of binomially distributed random variables will always be less than or equal to the variance of a binomial variable with the sa
Optional stopping theorem
In probability theory, the optional stopping theorem (or Doob's optional sampling theorem) says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its in
Stein's lemma
Stein's lemma, named in honor of Charles Stein, is a theorem of probability theory that is of interest primarily because of its applications to statistical inference — in particular, to James–Stein es
Isserlis' theorem
In probability theory, Isserlis' theorem or Wick's probability theorem is a formula that allows one to compute higher-order moments of the multivariate normal distribution in terms of its covariance m
Itô's lemma
In mathematics, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in French literature) is an identity used in Itô calculus to find the differential of a time-dependent fun
Gordon–Newell theorem
In queueing theory, a discipline within the mathematical theory of probability, the Gordon–Newell theorem is an extension of Jackson's theorem from open queueing networks to closed queueing networks o
Central limit theorem
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distri
Krylov–Bogolyubov theorem
In mathematics, the Krylov–Bogolyubov theorem (also known as the existence of invariant measures theorem) may refer to either of the two related fundamental theorems within the theory of dynamical sys
Reversed compound agent theorem
In probability theory, the reversed compound agent theorem (RCAT) is a set of sufficient conditions for a stochastic process expressed in any formalism to have a product form stationary distribution (
Hadwiger's theorem
In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem characterises the valuations on convex bodies in It was proved by Hugo Hadwiger.
Lovász local lemma
In probability theory, if a large number of events are all independent of one another and each has probability less than 1, then there is a positive (possibly small) probability that none of the event
Glivenko–Cantelli theorem
In the theory of probability, the Glivenko–Cantelli theorem (sometimes referred to as the Fundamental Theorem of Statistics), named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, determ
Arrival theorem
In queueing theory, a discipline within the mathematical theory of probability, the arrival theorem (also referred to as the random observer property, ROP or job observer property) states that "upon a
Reversible diffusion
In mathematics, a reversible diffusion is a specific example of a reversible stochastic process. Reversible diffusions have an elegant characterization due to the Russian mathematician Andrey Nikolaev
Marcinkiewicz–Zygmund inequality
In mathematics, the Marcinkiewicz–Zygmund inequality, named after Józef Marcinkiewicz and Antoni Zygmund, gives relations between moments of a collection of independent random variables. It is a gener
Poisson scatter theorem
In probability theory, The Poisson scatter theorem describes a probability model of . It implies that the number of points in a fixed region will follow a Poisson distribution.
Doob's martingale convergence theorems
In mathematics – specifically, in the theory of stochastic processes – Doob's martingale convergence theorems are a collection of results on the limits of supermartingales, named after the American ma
Freidlin–Wentzell theorem
In mathematics, the Freidlin–Wentzell theorem (due to Mark Freidlin and Alexander D. Wentzell) is a result in the large deviations theory of stochastic processes. Roughly speaking, the Freidlin–Wentze
Skorokhod's representation theorem
In mathematics and statistics, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence of probability measures whose limit measure is sufficiently well-behaved can
Chung–Erdős inequality
In probability theory, the Chung–Erdős inequality provides a lower bound on the probability that one out of many (possibly dependent) events occurs. The lower bound is expressed in terms of the probab
Campbell's theorem (probability)
In probability theory and statistics, Campbell's theorem or the Campbell–Hardy theorem is either a particular equation or set of results relating to the expectation of a function summed over a point p
Hewitt–Savage zero–one law
The Hewitt–Savage zero–one law is a theorem in probability theory, similar to Kolmogorov's zero–one law and the Borel–Cantelli lemma, that specifies that a certain type of event will either almost sur
Lévy's continuity theorem
In probability theory, Lévy’s continuity theorem, or Lévy's convergence theorem, named after the French mathematician Paul Lévy, connects convergence in distribution of the sequence of random variable
Isolation lemma
In theoretical computer science, the term isolation lemma (or isolating lemma) refers to randomized algorithms that reduce the number of solutions to a problem to one, should a solution exist.This is
Donsker's theorem
In probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the centra
Feller's coin-tossing constants
Feller's coin-tossing constants are a set of numerical constants which describe asymptotic probabilities that in n independent tosses of a fair coin, no run of k consecutive heads (or, equally, tails)
Dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the L1 norm. Its
Characterization of probability distributions
In mathematics in general, a characterization theorem says that a particular object – a function, a space, etc. – is the only one that possesses properties specified in the theorem. A characterization
Kunita–Watanabe inequality
In stochastic calculus, the Kunita–Watanabe inequality is a generalization of the Cauchy–Schwarz inequality to integrals of stochastic processes.It was first obtained by Hiroshi Kunita and Shinzo Wata