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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Lévy metric is a
metric Metric or metrical may refer to: Measuring * 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 ...
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. Ever ...
s of one-dimensional
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s. It is a special case of the Lévy–Prokhorov metric, 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. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/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 In mathematics, a càdlàg (), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset of them) that is everywhere right-continuous an ...
* Lévy–Prokhorov metric *
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 ...


References

* Measure theory Metric geometry Theory of probability distributions Paul Lévy (mathematician) {{metric-geometry-stub