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Uniform Convergence In Probability
Uniform convergence in probability is a form of convergence in probability in statistical asymptotic theory and probability theory. It means that, under certain conditions, the ''empirical frequencies'' of all events in a certain event-family converge to their ''theoretical probabilities''. Uniform convergence in probability has applications to statistics as well as machine learning as part of statistical learning theory. The law of large numbers says that, for each ''single'' event A, its empirical frequency in a sequence of independent trials converges (with high probability) to its theoretical probability. In many application however, the need arises to judge simultaneously the probabilities of events of an entire class S from one and the same sample. Moreover it, is required that the relative frequency of the events converge to the probability uniformly over the entire class of events S The Uniform Convergence Theorem gives a sufficient condition for this convergence to hol ... [...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 rando ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |