Lévy metric

2020 Mathematics Subject Classification: Primary: 60E05 [MSN][ZBL]

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) MR0428382 Zbl 0358.60020

Comments

A word of warning. In the Soviet mathematical literature (and in the main article above), distribution functions are usually left continuous, whereas in the West they are right continuous. So slight changes must be made in 2) or 7).

Let be a distribution function or, more generally, a non-decreasing left-continuous function. Then has a countable set of discontinuity points. The complement of this set is called the continuity set of . A series of distribution functions is said to converge weakly to a distribution if this is the case on the continuity set of . The series converges completely if moreover and . Cf. also Convergence of distributions and Convergence, types of.

References

[a1]	P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201
[a2]	W. Hengartner, R. Theodorescu, "Concentration functions" , Acad. Press (1973)
[a3]	M. Loève, "Probability theory" , v. Nostrand (1963) pp. 178 MR0203748 Zbl 0108.14202