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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Lévy metric is a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
on the space of
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...
s of one-dimensional
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s. It is a special case of the
Lévy–Prokhorov metric In mathematics, the Lévy–Prokhorov metric (sometimes known just as the Prokhorov metric) is a metric (mathematics), metric (i.e., a definition of distance) on the collection of probability measures on a given metric space. It is named after the F ...
, and is named after the French mathematician Paul Lévy.


Definition

Let F, G : \mathbb \to
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
/math> be two cumulative distribution functions. Define the Lévy distance between them to be :L(F, G) := \inf \. Intuitively, if between the graphs of ''F'' and ''G'' one inscribes squares with sides parallel to the coordinate axes (at points of discontinuity of a graph vertical segments are added), then the side-length of the largest such square is equal to ''L''(''F'', ''G''). A sequence of cumulative distribution functions \_^\infty weakly converges to another cumulative distribution function F if and only if L(F_n,F) \to 0.


See also

* Càdlàg *
Lévy–Prokhorov metric In mathematics, the Lévy–Prokhorov metric (sometimes known just as the Prokhorov metric) is a metric (mathematics), metric (i.e., a definition of distance) on the collection of probability measures on a given metric space. It is named after the F ...
*
Wasserstein metric In mathematics, the Wasserstein distance or Kantorovich– Rubinstein metric is a distance function defined between probability distributions on a given metric space M. It is named after Leonid Vaseršteĭn. Intuitively, if each distribution is ...


References

* {{DEFAULTSORT:Levy metric Measure theory Metric geometry Theory of probability distributions Paul Lévy (mathematician)