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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.
In mathematics Lévy's constant (sometimes known as the Khinchin–Lévy constant) occurs in an expression for the asymptotic behaviour of the denominators of the convergents of simple continued fractions. [1]
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.
The continuous mapping theorem states that for a continuous function g, if the sequence {X n} converges in distribution to X, then {g(X n)} converges in distribution to g(X). Note however that convergence in distribution of {X n} to X and {Y n} to Y does in general not imply convergence in distribution of {X n + Y n} to X + Y or of {X n Y n} to XY.
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. [note 1] It is a special case of the inverse-gamma distribution.
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which displacements in pairwise disjoint time intervals are independent, and displacements in different time intervals of the same length have identical ...
In mathematics and statistics, the continuity theorem may refer to one of the following results: the Lévy continuity theorem on random variables;
Continuity theorem may refer to one of two results: Lévy's continuity theorem, on random variables; Kolmogorov continuity theorem, on stochastic processes; In geometry: Parametric continuity, for parametrised curves; Geometric continuity, a concept primarily applied to the conic sections and related shapes; In probability theory