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Komlós–Major–Tusnády Approximation
In probability theory, the Komlós–Major–Tusnády approximation (also known as the KMT approximation, the KMT embedding, or the Hungarian embedding) refers to one of the two strong embedding theorems: 1) approximation of random walk by a standard Brownian motion constructed on the same probability space, and 2) an approximation of the empirical process by a Brownian bridge constructed on the same probability space. It is named after Hungarian mathematicians János Komlós (mathematician), János Komlós, Gábor Tusnády, and Péter Major, who proved it in 1975. Theory Let U_1,U_2,\ldots be independent Uniform distribution (continuous), uniform (0,1) random variables. Define a uniform empirical distribution function as :F_(t)=\frac\sum_^n \mathbf_,\quad t\in [0,1]. Define a uniform empirical process as :\alpha_(t)=\sqrt(F_(t)-t),\quad t\in [0,1]. The Donsker theorem (1952) shows that \alpha_(t) convergence of random variables, converges in law to a Brownian bridge B(t). Komlós, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory or probability calculus 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 of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), 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 (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
