Lecture 1: Uniform Integrability, Probabilistic Metrics

PREVIOUS SECTION: Coupling Inequality, Maximal Coupling, Dobrushin's Theorem

1.6 Probabilistic Metrics

The Dobrushin's theorem gives a positive solution of one of the important problems that arise in the theory of probabilistic metrics. Let us describe briefly some concepts of this theory. - complete separable metric space,

is a -algebra in , generated by all sets , ,

Problem No.4. Prove:

Let be any probability distribution on . Denote by its first marginal distribution, and by - second one:

Let be the set of all probability distributions (measures) on .

Definition 3

A function is called a probabilistic metric, if it satisfies:

(1) ;

(2) ;

(3)

has marginals and

(4) "triangle inequiality":

has marginals and
has marginals and

has marginals and

Definition 4

A probabilistic metric is simple if it depends on marginal distributions only
(i.e. if and have the same marginals, then ),
and complex - otherwise.

For simple metric, it is natural to write instead of , so is some "distance" between and .

For complex metric, we can write instead of , where is a coupling of two r.v.'s with joint distribution :

So, may be considered as a "distance" between r.v.'s.

We can also write for simple metrics.

Examples.

Simple	Complex
1) (Total variation norm (T.V.N.))	2) (Indicator metric (I.M.))
For real-valued r.v.'s:
3) (Uniform metric (U.M.))	5) (Ki Fan metric (K.F.M.))
4) (Levy metric (L.M.))