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Blumenthal's Zero–one Law
In the mathematical theory of probability, Blumenthal's zero–one law, named after Robert McCallum Blumenthal, is a statement about the nature of the beginnings of right continuous Feller process. Loosely, it states that any right continuous Feller process on [0,\infty) starting from deterministic point has also deterministic initial movement. Statement Suppose that X=(X_t:t\geq 0) is an adapted right continuous Feller process on a probability space (\Omega,\mathcal,\_,\mathbb) such that X_0 is constant with probability one. Let \mathcal^X_t:=\sigma(X_s; s\leq t), \mathcal^X_:=\bigcap_\mathcal^X_s. Then any event in the germ sigma algebra \Lambda \in \mathcal^X_ has either \mathbb(\Lambda)=0 or \mathbb(\Lambda)=1. Generalization Suppose that X=(X_t:t\geq 0) is an adapted stochastic process on a probability space (\Omega,\mathcal,\_,\mathbb) such that X_0 is constant with probability one. If X has Markov property with respect to the filtration \_ then any event \Lamb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero–one Law
In probability theory, a zero–one law is a result that states that an event must have probability 0 or 1 and no intermediate value. Sometimes, the statement is that the limit of certain probabilities must be 0 or 1. It may refer to: * Borel–Cantelli lemma * Blumenthal's zero–one law for Markov processes, * Engelbert–Schmidt zero–one law for continuous, nondecreasing additive functionals of Brownian motion, * Hewitt–Savage zero–one law for exchangeable sequences, * Kolmogorov's zero–one law for the tail σ-algebra, * Lévy's zero–one law, related to martingale convergence. * Topological zero–one law, related to meager set In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...s, * * Zero-one law (logic) for sentences valid in finite structures. {{DEFAULTSORT:Ze ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert McCallum Blumenthal
Robert "Bob" McCallum Blumenthal (7 February 1931, Chicago – 8 November 2012) was an American mathematician, specializing in probability theory. He is known for Blumenthal's zero-one law. Biography He received his Ph.D. in mathematics from Cornell University in 1956 under Gilbert Hunt with thesis ''An Extended Markov Property''. Blumenthal became in 1956 an instructor at the University of Washington, was eventually promoted to full professor, and in 1997 retired there. He was on sabbatical for the academic year 1961–1962 at the Institute for Advanced Study in Princeton and for the academic year 1966–1967 in Germany. Upon his death he was survived by his wife and two sons. Selected publications Articles * *with R. K. Getoor: *with R. K. Getoor: *with R. K. Getoor and Daniel Burrill Ray, D. B. Ray: *with R. K. Getoor: *with R. K. Getoor: Books *with R. K. Getoor: * References {{DEFAULTSORT:Blumenthal, Robert McCallum 1931 births 2012 deaths 20th-century ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Continuous
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feller Process
In probability theory relating to stochastic processes, a Feller process is a particular kind of Markov process. Definitions Let ''X'' be a locally compact Hausdorff space with a countable base. Let ''C''0(''X'') denote the space of all real-valued continuous functions on ''X'' that vanish at infinity, equipped with the sup-norm , , ''f'' , , . From analysis, we know that ''C''0(''X'') with the sup norm is a Banach space. A Feller semigroup on ''C''0(''X'') is a collection ''t'' ≥ 0 of positive linear maps from ''C''0(''X'') to itself such that * , , ''T''''t''''f'' , , ≤ , , ''f'' , , for all ''t'' ≥ 0 and ''f'' in ''C''0(''X''), i.e., it is a contraction (in the weak sense); * the semigroup property: ''T''''t'' + ''s'' = ''T''''t'' o''T''''s'' for all ''s'', ''t'' ≥ 0; * lim''t'' → 0, , ''T''''t''''f'' − ''f'' , , = 0 for every ''f'' in ''C''0('' ... [...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 sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Germ (mathematics)
In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind that captures their shared local properties. In particular, the objects in question are mostly functions (or maps) and subsets. In specific implementations of this idea, the functions or subsets in question will have some property, such as being analytic or smooth, but in general this is not needed (the functions in question need not even be continuous); it is however necessary that the space on/in which the object is defined is a topological space, in order that the word ''local'' has some meaning. Name The name is derived from ''cereal germ'' in a continuation of the sheaf metaphor, as a germ is (locally) the "heart" of a function, as it is for a grain. Formal definition Basic definition Given a point ''x'' of a topological space ''X'', and two maps f, g: X \to Y (where ''Y'' is any set), then f and g define the same germ at ''x'' if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sigma Algebra
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used as an operator for summation. When used at the end of a letter-case word (one that does not use all caps), the final form (ς) is used. In ' (Odysseus), for example, the two lowercase sigmas (σ) in the center of the name are distinct from the word-final sigma (ς) at the end. The Latin letter S derives from sigma while the Cyrillic letter Es derives from a lunate form of this letter. History The shape (Σς) and alphabetic position of sigma is derived from the Phoenician letter ( ''shin''). Sigma's original name may have been ''san'', but due to the complicated early history of the Greek epichoric alphabets, ''san'' came to be identified as a separate letter in the Greek alphabet, represented as Ϻ. Herodotus reports that "san" wa ... [...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] |
Markov Property
In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov property, except that the meaning of "present" is defined in terms of a random variable known as a stopping time. The term Markov assumption is used to describe a model where the Markov assumption is assumed to hold, such as a hidden Markov model. A Markov random field extends this property to two or more dimensions or to random variables defined for an interconnected network of items. An example of a model for such a field is the Ising model. A discrete-time stochastic process satisfying the Markov property is known as a Markov chain. Introduction A stochastic process has the Markov property if the conditional probability distribution of future states of the process (conditional on both past and present values) depends only upon the present ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Markov Property
In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov property, except that the meaning of "present" is defined in terms of a random variable known as a stopping time. The term Markov assumption is used to describe a model where the Markov assumption is assumed to hold, such as a hidden Markov model. A Markov random field extends this property to two or more dimensions or to random variables defined for an interconnected network of items. An example of a model for such a field is the Ising model. A discrete-time stochastic process satisfying the Markov property is known as a Markov chain. Introduction A stochastic process has the Markov property if the conditional probability distribution of future states of the process (conditional on both past and present values) depends only upon the present ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
