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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 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.
Lévy’s continuity theorem: The sequence {X n} converges in distribution to X if and only if the sequence of corresponding characteristic functions {φ n} converges pointwise to the characteristic function φ of X. Convergence in distribution is metrizable by the Lévy–Prokhorov metric.
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 ...
Download as PDF; Printable version; ... Lévy's modulus of continuity theorem ... This page was last edited on 7 April 2024, ...
Lévy's modulus of continuity theorem is a theorem that gives a result about an almost sure behaviour of an estimate of the modulus of continuity for Wiener process, that is used to model what's known as Brownian motion. Lévy's modulus of continuity theorem is named after the French mathematician Paul Lévy.
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 .
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.