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Borell–TIS Inequality
In mathematics and probability, the Borell–TIS inequality is a result bounding the probability of a deviation of the uniform norm of a centered Gaussian stochastic process above its expected value. The result is named for Christer Borell and its independent discoverers Boris Tsirelson, Ildar Ibragimov, and Vladimir Sudakov. The inequality has been described as "the single most important tool in the study of Gaussian processes." Statement Let T be a topological space, and let \_ be a centered (i.e. mean zero) Gaussian process on T, with ::\, f \, _T := \sup_ , f_t , almost surely finite, and let ::\sigma_T^2 := \sup_ \operatorname, f_t , ^2. Then \operatorname(\, f \, _T) and \sigma_T are both finite, and, for each u > 0, ::\operatorname \big( \, f \, _T > \operatorname(\, f \, _T) + u \big) \leq \exp\left( \frac \right). Another related statement which is also known as the Borell-TIS inequality is that, under the same conditions as above, ::\operatorname\big(\sup ... [...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 points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability
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] |
Uniform Norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when the supremum is in fact the maximum, the . The name "uniform norm" derives from the fact that a sequence of functions converges to under the metric derived from the uniform norm if and only if converges to uniformly. If is a continuous function on a closed and bounded interval, or more generally a compact set, then it is bounded and the supremum in the above definition is attained by the Weierstrass extreme value theorem, so we can replace the supremum by the maximum. In this case, the norm is also called the . In particular, if is some vector such that x = \left(x_1, x_2, \ldots, x_n\right) in finite dimensional coordinate space, it takes the form: :\, x\, _\infty := \max \left(\left, x_1\ , \ldots , \left, x_n\\right). Metric and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R''n'', closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the Germany, German mathematician Carl Friedrich Gauss. One reason why Gaussian measures are so ubiquitous in probability theory is the central limit theorem. Loosely speaking, it states that if a random variable ''X'' is obtained by summing a large number ''N'' of independent random variables of order 1, then ''X'' is of order \sqrt and its law is approximately Gaussian. Definitions Let ''n'' ∈ N and let ''B''0(R''n'') denote the complete measure, completion of the Borel sigma algebra, Borel ''σ''-algebra on R''n''. Let ''λ''''n'' : ''B''0(R''n'') → [0, +∞] denote the usual ''n''-dimensional Lebesgue measure. Then the standard Gaussian measure ''γ''''n'' : ''B''0(R''n'') → [0, 1] is defined by :\gamma^ (A) = \frac \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to end th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christer Borell
Christer or Krister are varieties of the masculine given name Kristian, derived from the Latin name ''Christianus'', which in turn comes from the Greek word ''khristianós'', which means "follower of Christ". The name, written in its two variants Christer and Krister, is quite common in the Nordic countries. Notable people with the name include: *Catherine Christer Hennix (born 1948), Swedish-American composer, philosopher, scientist and visual artist associated with drone minimal music * Christer Abris (formerly Abrahamsson, born 1947), Swedish former ice hockey goaltender * Christer Adelsbo, born 1962, is a Swedish social democratic politician who has been a member of the Riksdag since 2002 *Christer Basma (born 1972), Norwegian football coach and defender * Christer Björkman (born 1957), Swedish singer * Christer Boucht (1911–2009), Finnish-Swedish lawyer, adventure traveller and writer *Christer Boustedt (1939–1986), Swedish musician and actor *Christer Dahl (born 1940), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boris Tsirelson
Boris Semyonovich Tsirelson (May 4, 1950 – January 21, 2020) ( he, בוריס סמיונוביץ' צירלסון, russian: Борис Семёнович Цирельсон) was a Russian–Israeli mathematician and Professor of Mathematics at Tel Aviv University in Israel, as well as a Wikipedian, Wikipedia editor. Biography Tsirelson was born in Saint Petersburg, Leningrad to a History of the Jews in Russia, Russian Jewish family. From his father Simeon's side, he was the great-nephew of rabbi Yehuda Leib Tsirelson, chief rabbi of Bessarabia from 1918 to 1941, and a prominent posek and Jewish leader. He obtained his Master of Science from the Saint Petersburg State University, University of Leningrad and remained there to pursue graduate studies. He obtained his Doctor of Philosophy, Ph.D. in 1975, with thesis "General properties of bounded Gaussian processes and related questions" written under the direction of Ildar Abdulovich Ibragimov. Later, he participated in the Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ildar Ibragimov (mathematician)
Ildar Abdulovich Ibragimov (Ильдар Абдулович Ибрагимов, born 15 July 1932, Leningrad) is a Russian mathematician, specializing in probability theory and mathematical statistics. Biography Ibragimov is the son of a father who was an engineer with Bashkir ancestry and a mother who was a physician from a Tatar family with origins in Kazan. Ildar Ibragimov studied at Leningrad State University, where he graduated in mathematics in 1956. He received in 1960 his Russian candidate degree (Ph.D.) under Yuri Linnik and in 1967 his Russian doctorate (higher doctoral degree). In 1969 he became a professor of probability at Leningrad State University. He is a senior scientist and director of the laboratory of statistical methods at the Steklov Institute in Saint Petersburg, a position he has held there since 1972 as the successor to Yuri Linnik. Ibragimov was elected in 1990 a corresponding member and in 1997 a full member of the Russian Academy of Sciences. In 197 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Sudakov
Vladimir may refer to: Names * Vladimir (name) for the Bulgarian, Croatian, Czech, Macedonian, Romanian, Russian, Serbian, Slovak and Slovenian spellings of a Slavic name * Uladzimir for the Belarusian version of the name * Volodymyr for the Ukrainian version of the name * Włodzimierz (given name) for the Polish version of the name * Valdemar for the Germanic version of the name * Wladimir for an alternative spelling of the name Places * Vladimir, Russia, a city in Russia * Vladimir Oblast, a federal subject of Russia * Vladimir-Suzdal, a medieval principality * Vladimir, Ulcinj, a village in Ulcinj Municipality, Montenegro * Vladimir, Gorj, a commune in Gorj County, Romania * Vladimir, a village in Goiești Commune, Dolj County, Romania * Vladimir (river), a tributary of the Gilort in Gorj County, Romania * Volodymyr (city), a city in Ukraine Religious leaders * Metropolitan Vladimir (other), multiple * Jovan Vladimir (d. 1016), ruler of Doclea and a saint of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. The concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the Gaussian distribution (normal distribution). Gaussian processes can be seen as an infinite-dimensional generalization of multivariate normal distributions. Gaussian processes are useful in statistical modelling, benefiting from properties inherited from the normal distribution. For example, if a random process is modelled as a Gaussian process, the distribution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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