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In probability theory, Lévy’s continuity theorem, or Lévy's convergence theorem, [1] named after the French mathematician Paul Lévy, connects convergence in distribution of the sequence of random variables with pointwise convergence of their characteristic functions.
The characteristic function approach is particularly useful in analysis of linear combinations of independent random variables: a classical proof of the Central Limit Theorem uses characteristic functions and Lévy's continuity theorem. Another important application is to the theory of the decomposability of random variables.
In mathematics and statistics, the continuity theorem may refer to one of the following results: the Lévy continuity theorem on random variables ; the Kolmogorov continuity theorem on stochastic processes .
Continuous stochastic process: the question of continuity of a stochastic process is essentially a question of convergence, and many of the same concepts and relationships used above apply to the continuity question. Asymptotic distribution; Big O in probability notation; Skorokhod's representation theorem; The Tweedie convergence theorem ...
In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable.In spectroscopy, this distribution, with frequency as the dependent variable, is known as a van der Waals profile.
Martingale central limit theorem; Infinite divisibility (probability) Method of moments (probability theory) Stability (probability) Stein's lemma; Characteristic function (probability theory) Lévy continuity theorem; Darmois–Skitovich theorem; Edgeworth series; Helly–Bray theorem; Kac–Bernstein theorem; Location parameter; Maxwell's theorem
This theorem makes rigorous the intuitive notion of probability as the expected long-run relative frequency of an event's occurrence. It is a special case of any of several more general laws of large numbers in probability theory. Chebyshev's inequality. Let X be a random variable with finite expected value μ and finite non-zero variance σ 2.
Lévy was born in Paris to a Jewish family which already included several mathematicians. [3] His father Lucien Lévy was an examiner at the École Polytechnique.Lévy attended the École Polytechnique and published his first paper in 1905, at the age of nineteen, while still an undergraduate, in which he introduced the Lévy–Steinitz theorem.