PREVIOUS SECTION: Properties of UI
In this section, we assume that random variables are not necessary real-valued, but may take values in some measurable space that is assumed to be complete separable metric space.
Coupling inequality
Let be two -valued r.v.'s.
Put
Then, for ,
Therefore, for any , , that is
Maximal coupling
Let's reformulate the statement. Note that l.h.s. of inequality (*) depends on "marginal" distributions and only and does not depend on the joint distribution of and . Therefore, we get the following:
for given and and for any their coupling (*) takes place. Or, equivalently,
(?) May be, in is equality?
(??) If "yes", then does there exists such a coupling that
Both answers are positive! And this is the statement of Dobrushin's theorem.
Proof. is a signed measure. Therefore, Banach theorem states that
there exists a subset :
(a) ;
(b) .
Note:
1) if , then and the coupling is obvious;
2) .
Assume . Introduce 4 distributions (probability measures):
Similarly,
Then, define 5 independent r.v.'s: and
1 | 2 | 0 | |
Now we can "construct" and :
Simple calculations show that , .
Problem No.3. "Indeed, . . ." |
Then,
So,
and the proof is completed.
QDE
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