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Talagrand's Concentration Inequality
In the probability theory field of mathematics , Talagrand's concentration inequality is an isoperimetric-type inequality for product probability spaces. It was first proved by the French mathematician Michel Talagrand. The inequality is one of the manifestations of the concentration of measure phenomenon. Statement The inequality states that if \Omega = \Omega_1 \times \Omega_2 \times \cdots \times \Omega_n is a product space endowed with a product probability measure and A is a subset in this space, then for any t \ge 0 : \Pr \cdot \Pr\leftright Rights are legal, social, or ethical principles of freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal system, social convention, or ethical ...\le e^ \, , where is the complement of ''A_'' where this is defined by :A_t = \ and where \rho is Talagrand's convex distance defined as : \rho(A,x) = \max_ \ \min_ \ \sum_ \alpha_i w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isoperimetric
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n by its volume \operatorname(S), :\operatorname(S)\geq n \operatorname(S)^ \, \operatorname(B_1)^, where B_1\subset\R^n is a unit sphere. The equality holds only when S is a sphere in \R^n. On a plane, i.e. when n=2, the isoperimetric inequality relates the square of the circumference of a closed curve and the area of a plane region it encloses. '' Isoperimetric'' literally means "having the same perimeter". Specifically in \R ^2, the isoperimetric inequality states, for the length ''L'' of a closed curve and the area ''A'' of the planar region that it encloses, that : L^2 \ge 4\pi A, and that equality holds if and only if the curve is a circle. The isoperimetric problem is to determine a plane figure of the largest possible area ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequality (mathematics)
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities: * The notation ''a'' ''b'' means that ''a'' is greater than ''b''. In either case, ''a'' is not equal to ''b''. These relations are known as strict inequalities, meaning that ''a'' is strictly less than or strictly greater than ''b''. Equivalence is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: * The notation ''a'' ≤ ''b'' or ''a'' ⩽ ''b'' means that ''a'' is less than or equal to ''b'' (or, equivalently, at most ''b'', or not greater than ''b''). * The notation ''a'' ≥ ''b'' or ''a'' ⩾ ''b'' means that ''a'' is greater than or equal to ''b'' (or, equivalently, at least ''b'', or not less than ''b''). The r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Space
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product. Definition Throughout, I will be some non-empty index set and for every index i \in I, let X_i be a topological space. Denote the Cartesian product of the sets X_i by X := \prod X_ := \prod_ X_i and for every index i \in I, denote the i-th by \begin p_i :\;&& \prod_ X_j &&\;\to\; & X_i \\ .3ex && \left(x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a die. A probability space consists of three elements:Stroock, D. W. (1999). Probability theory: an analytic view. Cambridge University Press. # A sample space, \Omega, which is the set of all possible outcomes. # An event space, which is a set of events \mathcal, an event being a set of outcomes in the sample space. # A probability function, which assigns each event in the event space a probability, which is a number between 0 and 1. In order to provide a sensible model of probability, these elements must satisfy a number of axioms, detailed in this article. In the example of the throw of a standard die, we would take the sample space to be \. For the event space, we could simply use the set of all subsets of the sam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michel Talagrand
Michel Pierre Talagrand (born 15 February 1952) is a French mathematician. Docteur ès sciences since 1977, he has been, since 1985, Directeur de Recherches at CNRS and a member of the Functional Analysis Team of the Institut de Mathématique of Paris. Talagrand was elected as correspondent of the Académie des sciences of Paris in March 1997, and then as a full member in November 2004, in the Mathematics section. Talagrand studies mainly functional analysis and probability theory and their applications. Scientific activity Talagrand has been interested in probability with minimal structure. He has obtained a complete characterization of bounded Gaussian processes in very general settings, and also new methods to bound stochastic processes. He discovered new aspects of the isoperimetric and concentration of measure phenomena for product spaces, by obtaining inequalities which make use of new kind of distances between a point and a subset of a product space. These inequali ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concentration Of Measure
In mathematics, concentration of measure (about a median) is a principle that is applied in measure theory, probability and combinatorics, and has consequences for other fields such as Banach space theory. Informally, it states that "A random variable that depends in a Lipschitz way on many independent variables (but not too much on any of them) is essentially constant". The concentration of measure phenomenon was put forth in the early 1970s by Vitali Milman in his works on the local theory of Banach spaces, extending an idea going back to the work of Paul Lévy. It was further developed in the works of Milman and Gromov, Maurey, Pisier, Schechtman, Talagrand, Ledoux, and others. The general setting Let (X, d) be a metric space with a measure \mu on the Borel sets with \mu(X) = 1. Let :\alpha(\epsilon) = \sup \left\, where :A_\epsilon = \left\ is the \epsilon-''extension'' (also called \epsilon-fattening in the context of the Hausdorff distance) of a set A. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Measure
In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of two topological spaces, except that there can be many natural choices for the product measure. Let (X_1, \Sigma_1) and (X_2, \Sigma_2) be two measurable spaces, that is, \Sigma_1 and \Sigma_2 are sigma algebras on X_1 and X_2 respectively, and let \mu_1 and \mu_2 be measures on these spaces. Denote by \Sigma_1 \otimes \Sigma_2 the sigma algebra on the Cartesian product X_1 \times X_2 generated by subsets of the form B_1 \times B_2, where B_1 \in \Sigma_1 and B_2 \in \Sigma_2. This sigma algebra is called the ''tensor-product σ-algebra'' on the product space. A ''product measure'' \mu_1 \times \mu_2 (also denoted by \mu_1 \otimes \mu_2 by many authors) is defined to be a measure on the measurable space (X_1 \times X_2, \Sigma_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probabilistic Inequalities
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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