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Martingale Convergence Theorem
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 bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lp Space
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki group they were first introduced by Frigyes Riesz . spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. Preliminaries The -norm in finite dimensions The Euclidean length of a vector x = (x_1, x_2, \dots, x_n) in the n-dimensional real vector space \Reals^n is given by the Euclidean norm: \, x\, _2 = \left(^2 + ^2 + \dotsb + ^2\right)^. The Euclidean distance between two points x and y is the length \, x - y\, _2 of the straight line b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sigma Algebra
Sigma ( ; uppercase Σ, lowercase σ, lowercase in word-final position ς; ) 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" was the name g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filtration (probability Theory)
In the theory of stochastic processes, a subdiscipline of probability theory, filtrations are totally ordered collections of subsets that are used to model the information that is available at a given point and therefore play an important role in the formalization of random (stochastic) processes. Definition Let (\Omega, \mathcal A, P) be a probability space and let I be an index set with a total order \leq (often \N , \R^+ , or a subset of \mathbb R^+ ). For every i \in I let \mathcal F_i be a sub-''σ''-algebra of \mathcal A . Then : \mathbb F:= (\mathcal F_i)_ is called a filtration, if \mathcal F_k \subseteq \mathcal F_\ell for all k \leq \ell . So filtrations are families of ''σ''-algebras that are ordered non-decreasingly. If \mathbb F is a filtration, then (\Omega, \mathcal A, \mathbb F, P) is called a filtered probability space. Example Let (X_n)_ be a stochastic process In probability theory and related fields, a stochastic () or rand ... [...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 . 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 of a random process under consideration. # An event space, \mathcal, which is a set of events, where an event is a subset of outcomes in the sample space. # A '' probability function'', P, which assigns, to each event in the event space, a probability, which is a number between 0 and 1 (inclusive). In order to provide a model of probability, these elements must satisfy probability axioms. In the example of the throw of a standard die, # The sample space \Omega is typically the set \ where each element in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Expectation
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on only a finite number of values, the "conditions" are that the variable can only take on a subset of those values. More formally, in the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of a set, partition of this probability space. Depending on the context, the conditional expectation can be either a random variable or a function. The random variable is denoted E(X\mid Y) analogously to conditional probability. The function form is either denoted E(X\mid Y=y) or a separate function symbol such as f(y) is introduced with the meaning E(X\mid Y) = f(Y). Examples Example 1: Dice rolling Consider the roll of a fair die and let ''A'' = 1 if the number is even (i.e., 2, 4, or 6) and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sample Continuous Process
In mathematics, a sample-continuous process is a stochastic process whose sample paths are almost surely continuous functions. Definition Let (Ω, Σ, P) be a probability space. Let ''X'' : ''I'' × Ω → ''S'' be a stochastic process, where the index set ''I'' and state space ''S'' are both topological spaces. Then the process ''X'' is called sample-continuous (or almost surely continuous, or simply continuous) if the map ''X''(''ω'') : ''I'' → ''S'' is continuous as a function of topological spaces for P-almost all ''ω'' in ''Ω''. In many examples, the index set ''I'' is an interval of time, , ''T''or real_line.html" ;"title=", +∞), and the state space ''S'' is the real line">, +∞), and the state space ''S'' is the real line or ''n''-dimensional Euclidean space R''n''. Examples * Brownian motion (the Wiener process) on Euclidean space is samp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filtration (mathematics)
In mathematics, a filtration \mathcal is, informally, like a set of ever larger Russian dolls, each one containing the previous ones, where a "doll" is a subobject of an algebraic structure. Formally, a filtration is an indexed family (S_i)_ of subobjects of a given algebraic structure S, with the index i running over some totally ordered index set I, subject to the condition that ::if i\leq j in I, then S_i\subseteq S_j. If the index i is the time parameter of some stochastic process, then the filtration can be interpreted as representing all historical but not future information available about the stochastic process, with the algebraic structure S_i gaining in complexity with time. Hence, a process that is adapted to a filtration \mathcal is also called non-anticipating, because it cannot "see into the future". Sometimes, as in a filtered algebra, there is instead the requirement that the S_i be subalgebras with respect to some operations (say, vector addition), but n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Sketch
Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a construct in proof theory * Mathematical proof, a convincing demonstration that some mathematical statement is necessarily true * Proof complexity, computational resources required to prove statements * Proof procedure, method for producing proofs in proof theory * Proof theory, a branch of mathematical logic that represents proofs as formal mathematical objects * Statistical proof, demonstration of degree of certainty for a hypothesis Law and philosophy * Evidence, information which tends to determine or demonstrate the truth of a proposition * Evidence (law), tested evidence or a legal proof * Legal burden of proof, duty to establish the truth of facts in a trial * Philosophic burden of proof, obligation on a party in a dispute to provide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur has probability 0, even though the set might not be empty. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space with a non-zero probability for each outcome, there is no difference between ''almost surely'' and ''surely'' (since having a probability of 1 entails including all the sample points); however, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, the continuity of the paths of Brownian motion, and the infinite monkey theorem. The terms almost certai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. the other being Derivative, differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter. A definite integral computes the signed area of the region in the plane that is bounded by the Graph of a function, graph of a given Function (mathematics), function between two points in the real line. Conventionally, areas above the horizontal Coordinate axis, axis of the plane are positive while areas below are n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
