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Sazonov's Theorem
In mathematics, Sazonov's theorem, named after Vyacheslav Vasilievich Sazonov (), is a theorem in functional analysis. It states that a bounded linear operator between two Hilbert spaces is ''γ''-radonifying if it is a Hilbert–Schmidt operator. The result is also important in the study of stochastic processes and the Malliavin calculus, since results concerning probability measures on infinite-dimensional spaces are of central importance in these fields. Sazonov's theorem also has a converse: if the map is not Hilbert–Schmidt, then it is not ''γ''-radonifying. Statement of the theorem Let ''G'' and ''H'' be two Hilbert spaces and let ''T'' : ''G'' → ''H'' be a bounded operator from ''G'' to ''H''. Recall that ''T'' is said to be ''γ''-radonifying if the push forward of the canonical Gaussian cylinder set measure on ''G'' is a ''bona fide'' measure on ''H''. Recall also that ''T'' is said to be a Hilbert–Schmidt operator if there is an orthonormal basis of ' ... [...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] |
Pushforward Measure
In measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a measure from one measurable space to another using a measurable function. Definition Given measurable spaces (X_1,\Sigma_1) and (X_2,\Sigma_2), a measurable mapping f\colon X_1\to X_2 and a measure \mu\colon\Sigma_1\to ,+\infty/math>, the pushforward of \mu is defined to be the measure f_(\mu)\colon\Sigma_2\to ,+\infty/math> given by :f_ (\mu) (B) = \mu \left( f^ (B) \right) for B \in \Sigma_. This definition applies ''mutatis mutandis'' for a signed or complex measure. The pushforward measure is also denoted as \mu \circ f^, f_\sharp \mu, f \sharp \mu, or f \# \mu. Main property: change-of-variables formula Theorem:Sections 3.6–3.7 in A measurable function ''g'' on ''X''2 is integrable with respect to the pushforward measure ''f''∗(''μ'') if and only if the composition g \circ f is integrable with respect to the measure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Processes
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 Browni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Function
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when is the identity function, the equality is true for all values of to which can be applied. Definition Formally, if is a set, the identity function on is defined to be a function with as its domain and codomain, satisfying In other words, the function value in the codomain is always the same as the input element in the domain . The identity function on is clearly an injective function as well as a surjective function, so it is bijective. The identity function on is often denoted by . In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or ''diagonal'' of . Algebraic properties If is any function, then we h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cylinder Set Measure
In mathematics, cylinder set measure (or promeasure, or premeasure, or quasi-measure, or CSM) is a kind of prototype for a measure on an infinite-dimensional vector space. An example is the Gaussian cylinder set measure on Hilbert space. Cylinder set measures are in general not measures (and in particular need not be countably additive but only finitely additive), but can be used to define measures, such as classical Wiener measure on the set of continuous paths starting at the origin in Euclidean space. Definition Let E be a separable real topological vector space. Let \mathcal (E) denote the collection of all surjective continuous linear maps T : E \to F_T defined on E whose image is some finite-dimensional real vector space F_T: \mathcal (E) := \left\. A cylinder set measure on E is a collection of probability measures \left\. where \mu_T is a probability measure on F_T. These measures are required to satisfy the following consistency condition: if \pi_ : F_S \to F_T is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prokhorov's Theorem
In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilyevich Prokhorov, who considered probability measures on complete separable metric spaces. The term "Prokhorov’s theorem" is also applied to later generalizations to either the direct or the inverse statements. Statement Let (S, \rho) be a separable metric space. Let \mathcal(S) denote the collection of all probability measures defined on S (with its Borel σ-algebra). Theorem. # A collection K\subset \mathcal(S) of probability measures is tight if and only if the closure of K is sequentially compact in the space \mathcal(S) equipped with the topology of weak convergence. # The space \mathcal(S) with the topology of weak convergence is metrizable. # Suppose that in addition, (S,\rho) is a complete metric space (so that (S,\rho) is a Polish space) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthonormal Basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space \R^n is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for \R^n arises in this fashion. For a general inner product space V, an orthonormal basis can be used to define normalized orthogonal coordinates on V. Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of \R^n under dot product. Every finite-dimensional inner product space has an orthonormal basis, wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cylinder Set Measure
In mathematics, cylinder set measure (or promeasure, or premeasure, or quasi-measure, or CSM) is a kind of prototype for a measure on an infinite-dimensional vector space. An example is the Gaussian cylinder set measure on Hilbert space. Cylinder set measures are in general not measures (and in particular need not be countably additive but only finitely additive), but can be used to define measures, such as classical Wiener measure on the set of continuous paths starting at the origin in Euclidean space. Definition Let E be a separable real topological vector space. Let \mathcal (E) denote the collection of all surjective continuous linear maps T : E \to F_T defined on E whose image is some finite-dimensional real vector space F_T: \mathcal (E) := \left\. A cylinder set measure on E is a collection of probability measures \left\. where \mu_T is a probability measure on F_T. These measures are required to satisfy the following consistency condition: if \pi_ : F_S \to F_T is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events; for example, the value assigned to "1 or 2" in a throw of a dice should be the sum of the values assigned to "1" and "2". Probability measures have applications in diverse fields, from physics to finance and biology. Definition The requirements for a function \mu to be a probability measure on a probability space are that: * \mu must return results in the unit interval , 1 returning 0 for the empty set and 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vyacheslav Vasilievich Sazonov
Vyacheslav Vasilievich Sazonov (Вячеслав Васильевич Сазонов, born August 25, 1935, Moscow – February 3, 2002, Moscow) was a Soviet-Russian mathematician, specializing in probability and measure theory. He is known for Sazonov's theorem. Education and career In 1958 he graduated from Moscow State University. There he received in 1961 his Ph.D. under Yuri Prokhorov with thesis "Распределения вероятностей и характеристические функционалы" (Probability distributions and characteristic functionals). Sazonov worked in the Steklov Institute of Mathematics from 1958 to 2002. In 1968 he received his Russian doctorate of sciences (Doctor Nauk) with thesis "Исследования по многомерным и бесконечномерным предельным теоремам теории вероятностей" (Investigations of multidimensional, infinite-dimensional and limit theorems of the theory of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Malliavin Calculus
In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows the computation of derivatives of random variables. Malliavin calculus is also called the stochastic calculus of variations. P. Malliavin first initiated the calculus on infinite dimensional space. Then, the significant contributors such as S. Kusuoka, D. Stroock, Bismut, S. Watanabe, I. Shigekawa, and so on finally completed the foundations. Malliavin calculus is named after Paul Malliavin whose ideas led to a proof that Hörmander's condition implies the existence and smoothness of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations. The calculus has been applied to stochastic partial differential equations as well. The calculus allows ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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