# Lévy metric

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A metric in the space of distribution functions (cf. Distribution function) of one-dimensional random variables such that:

for any . It was introduced by P. Lévy (see [1]). If between the graphs of and one inscribes squares with sides parallel to the coordinate axes (at points of discontinuity of a graph vertical segments are added), then a side of the largest of them is equal to .

The Lévy metric can be regarded as a special case of the Lévy–Prokhorov metric. The definition of the Lévy metric carries over to the set of all non-decreasing functions on (infinite values of the metric being allowed).

## Most important properties of the Lévy metric.

1) The Lévy metric induces a weak topology in (cf. Distributions, convergence of). The metric space () is separable and complete. Convergence of a sequence of functions from in the metric is equivalent to complete convergence.

2) If and if

then for any ,

3) Regularity of the Lévy metric: For any ,

(here denotes convolution, cf. Convolution of functions). A consequence of this property is the property of semi-additivity:

and the "smoothing inequality" :

( being a distribution that is degenerate at zero).

4) If , , then

5) If , , is an absolute moment of the distribution , then

6) The Lévy metric on is related to the integral mean metric

by the inequality

7) The Lévy metric on is related to the uniform metric

by the relations

 (*)

where

( is the concentration function for ). In particular, if one of the functions, for example , has a uniformly bounded derivative, then

A consequence of (*) is the Pólya–Glivenko theorem on the equivalence of weak and uniform convergence in the case when the limit distribution is continuous.

8) If , where and are constants, then for any ,

(in particular, the Lévy metric is invariant with respect to a shift of the distributions) and

9) If and are the characteristic functions (cf. Characteristic function) corresponding to the distributions and , then for any ,

The concept of the Lévy metric can be extended to the case of distributions in .

#### References

 [1] P. Lévy, "Théorie de l'addition des variables aléatoires" , Gauthier-Villars (1937) [2] V.M. Zolotarev, "Estimates of the difference between distributions in the Lévy metric" Proc. Steklov Inst. Math. , 112 (1973) pp. 232–240 Trudy Mat. Inst. Steklov. , 112 (1971) pp. 224–231 [3] V.M. Zolotarev, V.V. Senatov, "Two-sided estimates of Lévy's metric" Theor. Probab. Appl. , 20 (1975) pp. 234–245 Teor. Veroyatnost. i Primenen. , 20 : 2 (1975) pp. 239–250 [4] Yu.V. Linnik, I.V. Ostrovskii, "Decomposition of random variables and vectors" , Amer. Math. Soc. (1977) (Translated from Russian)