# Lévy metric

2010 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 [Le]). 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 .