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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] |
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. Typically these axioms formalise probability in terms of a probability space, which assigns a 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. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 gave statement to the lemma in the first decades of the 20th century. A related result, sometimes called the second Borel–Cantelli lemma, is a partial converse of the first Borel–Cantelli lemma. The lemma states that, under certain conditions, an event will have probability of either zero or one. Accordingly, it is the best-known of a class of similar theorems, known as zero-one laws. Other examples include Kolmogorov's zero–one law and the Hewitt–Savage zero–one law. Statement of lemma for probability spaces Let ''E''1,''E''2,... be a sequence of events in some probability space. The Borel–Cantelli lemma states: Here, "lim sup" denotes limit supremum of the sequence of events, and each event is a set of outcomes. That is, lim sup ''E''''n'' is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Markov Process
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs ''now''." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). It is named after the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics. Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 either 0 or 1, without the possibility of an intermediate value. This zero-one law is used in the study of questions of finiteness and asymptotic behavior for stochastic differential equations. (A Wiener process is a mathematical formalization of Brownian motion used in the statement of the theorem.) This 0-1 law, published in 1981, is named after Hans-Jürgen Engelbert and the probabilist Wolfgang Schmidt (not to be confused with the number theorist Wolfgang M. Schmidt). Engelbert–Schmidt 0–1 law Let \mathcal be a σ-algebra and let F = (\mathcal_t)_ be an increasing family of sub-''σ''-algebras of \mathcal. Let (W, F) be a Wiener process on the probability space (\Omega, \mathcal, P). Suppose that f is a Borel measurable function of the real line into ,∞ Then the following ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 surely happen or almost surely not happen. It is sometimes known as the Savage-Hewitt law for symmetric events. It is named after Edwin Hewitt and Leonard Jimmie Savage. Statement of the Hewitt-Savage zero-one law Let \left\_^\infty be a sequence of independent and identically-distributed random variables taking values in a set \mathbb. The Hewitt-Savage zero–one law says that any event whose occurrence or non-occurrence is determined by the values of these random variables and whose occurrence or non-occurrence is unchanged by finite permutations of the indices, has probability either 0 or 1 (a "finite" permutation is one that leaves all but finitely many of the indices fixed). Somewhat more abstractly, define the ''exchangeable sigma algebra'' or ''sigma algebra of symmetric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 almost surely happen or almost surely not happen; that is, the probability of such an event occurring is zero or one. Tail events are defined in terms of countably infinite families of σ-algebras. For illustrative purposes, we present here the special case in which each sigma algebra is generated by a random variable X_k for k\in\mathbb N. Let \mathcal be the sigma-algebra generated jointly by all of the X_k. Then, a tail event F \in \mathcal is an event which is probabilistically independent of each finite subset of these random variables. (Note: F belonging to \mathcal implies that membership in F is uniquely determined by the values of the X_k, but the latter condition is strictly weaker and does not suffice to prove the zero-one law.) For example, the event that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lévy's Zero–one Law
In mathematicsspecifically, in the theory of stochastic processesDoob's martingale convergence theorems are a collection of results on the limits of supermartingales, named after the American mathematician Joseph L. Doob. Informally, the martingale convergence theorem typically refers to the result that any supermartingale satisfying a certain boundedness condition must converge. One may think of supermartingales as the random variable analogues of non-increasing sequences; from this perspective, the martingale convergence theorem is a random variable analogue of the monotone convergence theorem, which states that any bounded monotone sequence converges. There are symmetric results for submartingales, which are analogous to non-decreasing sequences. Statement for discrete-time martingales A common formulation of the martingale convergence theorem for discrete-time martingales is the following. Let X_1, X_2, X_3, \dots be a supermartingale. Suppose that the supermartingale is b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 called nonmeagre, or of the second category. See below for definitions of other related terms. The meagre subsets of a fixed space form a σ-ideal of subsets; that is, any subset of a meagre set is meagre, and the union of countably many meagre sets is meagre. Meagre sets play an important role in the formulation of the notion of Baire space and of the Baire category theorem, which is used in the proof of several fundamental results of functional analysis. Definitions Throughout, X will be a topological space. A subset of X is called X, a of X, or of the in X if it is a countable union of nowhere dense subsets of X (where a nowhere dense set is a set whose closure has empty interior). The qualifier "in X" can be omitted if the ambient ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero-one Law (logic)
Finite model theory is a subarea of model theory. Model theory is the branch of logic which deals with the relation between a formal language (syntax) and its interpretations (semantics). Finite model theory is a restriction of model theory to interpretations on finite structures, which have a finite universe. Since many central theorems of model theory do not hold when restricted to finite structures, finite model theory is quite different from model theory in its methods of proof. Central results of classical model theory that fail for finite structures under finite model theory include the compactness theorem, Gödel's completeness theorem, and the method of ultraproducts for first-order logic (FO). While model theory has many applications to mathematical algebra, finite model theory became an "unusually effective" instrument in computer science. In other words: "In the history of mathematical logic most interest has concentrated on infinite structures. ..Yet, the objects co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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