# Lévy metric

2010 Mathematics Subject Classification: Primary: 60E05 [MSN][ZBL]

A metric $L$ in the space ${\mathcal F}$ of distribution functions (cf. Distribution function) of one-dimensional random variables such that:

$$L \equiv L ( F , G ) =$$

$$= \ \inf \{ \epsilon : {F ( x - \epsilon ) - \epsilon \leq G ( x) \leq F ( x + \epsilon ) + \epsilon \textrm{ for all } x } \}$$

for any $F , G \in {\mathcal F}$. It was introduced by P. Lévy (see [Le]). 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 a side of the largest of them is equal to $L$.

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 $M$ of all non-decreasing functions on $\mathbf R ^ {1}$( infinite values of the metric being allowed).

## Most important properties of the Lévy metric.

1) The Lévy metric induces a weak topology in ${\mathcal F}$( cf. Distributions, convergence of). The metric space ( ${\mathcal F} , L$) is separable and complete. Convergence of a sequence of functions from $M$ in the metric $L$ is equivalent to complete convergence.

2) If $F \in M$ and if

$$F _ {-} 1 ( x) = \inf \{ {t } : {F ( t) < x } \} ,$$

then for any $F , G \in M$,

$$L ( F , G ) = L ( F _ {-} 1 , G _ {-} 1 ) .$$

3) Regularity of the Lévy metric: For any $F , G , H \in {\mathcal F}$,

$$L ( F \star H , G \star H ) \leq L ( F , G )$$

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

$$L ( F _ {1} \star F _ {2} , G _ {1} \star G _ {2} ) \leq L ( F _ {1} ,\ G _ {1} ) + L ( F _ {2} , G _ {2} )$$

and the "smoothing inequality" :

$$L ( F , G ) \leq L ( F \star H , G \times H ) + 2L ( E , H )$$

( $E$ being a distribution that is degenerate at zero).

4) If $\alpha _ {k} \geq 0$, $F _ {k} , G _ {k} \in {\mathcal F}$, then

$$L \left ( \sum \alpha _ {k} F _ {k} , \sum \alpha _ {k} G _ {k} \right ) \leq \ \max \left ( 1 , \sum \alpha _ {k} \right ) \max L ( F _ {k} , G _ {k} ) .$$

5) If $\beta _ {r} ( F )$, $r > 0$, is an absolute moment of the distribution $F$, then

$$L ( F , E ) \leq \{ \beta _ {r} ( F ) \} ^ {r / ( r+ 1 ) } .$$

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

$$\rho _ {1} = \rho _ {1} ( F , G ) = \int\limits | F ( x) - G ( x) | dx$$

by the inequality

$$L ^ {2} \leq \rho _ {1} .$$

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

$$\rho = \rho ( F , G ) = \sup _ { x } | F ( x) - G ( x) |$$

by the relations

$$\tag{* } L \leq \rho \leq L + \min \{ Q _ {F} ( L) , Q _ {G} ( L) \} ,$$

where

$$Q _ {F} ( x) = \sup _ { t } | F ( t+ x ) - F ( t) |$$

( $Q _ {F} ( x)$ is the concentration function for $F \in {\mathcal F}$). In particular, if one of the functions, for example $G$, has a uniformly bounded derivative, then

$$\rho \leq \left ( 1 + \sup _ { x } G ^ \prime ( x) \right ) L .$$

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 $F _ {a , \sigma } ( x) = F ( \sigma x + a )$, where $a$ and $\sigma > 0$ are constants, then for any $F , G \in {\mathcal F}$,

$$L ( \sigma F , \sigma G ) \leq \sigma L ( F _ {a , \sigma } , G _ {a , \sigma } )$$

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

$$\lim\limits _ {\sigma \rightarrow 0 } L ( F _ {a , \sigma } , G _ {a , \sigma } ) = \rho ( F , G ) .$$

9) If $f$ and $g$ are the characteristic functions (cf. Characteristic function) corresponding to the distributions $F$ and $G$, then for any $T > e$,

$$L ( F , G ) \leq \frac{1} \pi \int\limits _ { 0 } ^ { T } | f ( t) - g ( t) | \frac{dt}{t} + 2e \frac{ \mathop{\rm ln} T }{T} .$$

The concept of the Lévy metric can be extended to the case of distributions in $\mathbf R ^ {n}$.

#### References

 [Le] P. Lévy, "Théorie de l'addition des variables aléatoires" , Gauthier-Villars (1937) [Z] 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 [ZS] 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 [LO] Yu.V. Linnik, I.V. Ostrovskii, "Decomposition of random variables and vectors" , Amer. Math. Soc. (1977) (Translated from Russian) MR0428382 Zbl 0358.60020

Let $F$ be a distribution function or, more generally, a non-decreasing left-continuous function. Then $F$ has a countable set of discontinuity points. The complement of this set is called the continuity set $C ( F )$ of $F$. A series of distribution functions $F _ {n}$ is said to converge weakly to a distribution $F$ if this is the case on the continuity set $C ( F )$ of $F$. The series converges completely if moreover $F _ {n} ( + \infty ) \rightarrow F ( \infty )$ and $F _ {n} ( - \infty ) \rightarrow F ( - \infty )$. Cf. also Convergence of distributions and Convergence, types of.