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Continuous Mapping Theorem
In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random variables. A continuous function, in Heine’s definition, is such a function that maps convergent sequences into convergent sequences: if ''xn'' → ''x'' then ''g''(''xn'') → ''g''(''x''). The ''continuous mapping theorem'' states that this will also be true if we replace the deterministic sequence with a sequence of random variables , and replace the standard notion of convergence of real numbers “→” with one of the types of convergence of random variables. This theorem was first proved by Henry Mann and Abraham Wald in 1943, and it is therefore sometimes called the Mann–Wald theorem. Meanwhile, Denis Sargan refers to it as the general transformation theorem. Statement Let , ''X'' be random elements defined on a metric space ''S''. Suppose a function (where ''S′'' is another metric space) has the set of discontinu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contraction Mapping Theorem
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points. It can be understood as an abstract formulation of Picard's method of successive approximations. The theorem is named after Stefan Banach (1892–1945) who first stated it in 1922. Statement ''Definition.'' Let (X, d) be a complete metric space. Then a map T : X \to X is called a contraction mapping on ''X'' if there exists q \in non-empty complete metric space with a contraction mapping T : X \to X. Then ''T'' admits a unique Fixed point (mathematics)">fixed-point x^* in ''X'' (i.e. T(x^*) = x^*). Furthermore, x^* can be found as follows: start with an arbitrary element x_0 \in X and define a sequence (x_n)_ by x_n = T(x_) for n \geq 1. Then \li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence In Distribution
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied. The different possible notions of convergence relate to how such a behavior can be characterized: two readily understood behaviors are that the sequence eventually takes a constant value, and that values in the sequence continue to change but can be described by an unchanging probability distribution. Background "Stochastic convergence" formalizes the idea that a sequence of essentially random or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theorems
Probability is the branch of mathematics concerning numerical descriptions of how likely an 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 as 0.5 or 50%). These conce ... [...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] |
Portmanteau Theorem
In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by ''convergence of measures'', consider a sequence of measures μ''n'' on a space, sharing a common collection of measurable sets. Such a sequence might represent an attempt to construct 'better and better' approximations to a desired measure μ that is difficult to obtain directly. The meaning of 'better and better' is subject to all the usual caveats for taking limits; for any error tolerance ε > 0 we require there be ''N'' sufficiently large for ''n'' ≥ ''N'' to ensure the 'difference' between μ''n'' and μ is smaller than ε. Various notions of convergence specify precisely what the word 'difference' should mean in that description; these notions are not equivalent to one another, and vary in strength. Three of the most common notions of convergence are described below. Informal descriptions This section attempts to pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slutsky's Theorem
In probability theory, Slutsky’s theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables. The theorem was named after Eugen Slutsky. Slutsky's theorem is also attributed to Harald Cramér. Statement Let X_n, Y_n be sequences of scalar/vector/matrix random elements. If X_n converges in distribution to a random element X and Y_n converges in probability to a constant c, then * X_n + Y_n \ \xrightarrow\ X + c ; * X_nY_n \ \xrightarrow\ Xc ; * X_n/Y_n \ \xrightarrow\ X/c, provided that ''c'' is invertible, where \xrightarrow denotes convergence in distribution. Notes: # The requirement that ''Yn'' converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid. For example, let X_n \sim (0,1) and Y_n = -X_n. The sum X_n + Y_n = 0 for all values of ''n''. Moreover, Y_n \, \xrightarrow \, (-1,0), but X_n + Y_n does not conv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Portmanteau Theorem
In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by ''convergence of measures'', consider a sequence of measures μ''n'' on a space, sharing a common collection of measurable sets. Such a sequence might represent an attempt to construct 'better and better' approximations to a desired measure μ that is difficult to obtain directly. The meaning of 'better and better' is subject to all the usual caveats for taking limits; for any error tolerance ε > 0 we require there be ''N'' sufficiently large for ''n'' ≥ ''N'' to ensure the 'difference' between μ''n'' and μ is smaller than ε. Various notions of convergence specify precisely what the word 'difference' should mean in that description; these notions are not equivalent to one another, and vary in strength. Three of the most common notions of convergence are described below. Informal descriptions This section attempts to pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Sure Convergence
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied. The different possible notions of convergence relate to how such a behavior can be characterized: two readily understood behaviors are that the sequence eventually takes a constant value, and that values in the sequence continue to change but can be described by an unchanging probability distribution. Background "Stochastic convergence" formalizes the idea that a sequence of essentially random or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence In Probability
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied. The different possible notions of convergence relate to how such a behavior can be characterized: two readily understood behaviors are that the sequence eventually takes a constant value, and that values in the sequence continue to change but can be described by an unchanging probability distribution. Background "Stochastic convergence" formalizes the idea that a sequence of essentially random or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discontinuity (mathematics)
Continuous functions are of utmost importance in mathematics, functions and applications. However, not all Function (mathematics), functions are Continuous function, continuous. If a function is not continuous at a point in its Domain of a function, domain, one says that it has a discontinuity there. The Set theory, set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function. This article describes the classification of discontinuities in the simplest case of functions of a single Real number, real Variable-sweep wing, variable taking real values. The Oscillation (mathematics), oscillation of a function at a point quantifies these discontinuities as follows: * in a removable discontinuity, the distance that the value of the function is off by is the oscillation; * in a jump discontinuity, the size of the jump is the oscillation (assuming that the value ''at'' the point lies between these limits of the two sides); * i ... [...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] |
Metric Space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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