PREVIOUS SECTION: Coupling Inequality, Maximal Coupling, Dobrushin's Theorem
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) ;
(4) "triangle inequiality":
|
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.)) |
One of the general problem in the theory of probabilistic metrics is:
Assume some simple metric to be given. Does there exist a complex metric such that
|
Theorem 1
|
The answer on the above question is positive for
the metrics: T.V.N. I.M. L.M. K.F.M. |
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