- for r.v.'s
- for r.v.'s
- distribution
function, 
- density
- probability
and probability measure, 
-
expectation, 
- variance
means 
for all 
means 
Standart families of distributions:
 |
 |
 |
 |
 |
 |
 - indicator
function. |
|
Convergence:
means 
means 
as 
Weak convergence:
,
if for each
such that 
is continuous in
,
Equivalent form:
, if for each 
- bounded continuous,
I will write also: 
. It
means: 
, and .
is a copy of 
they have the same distribution 
. In general, and may be defined on different probability spaces.
Coupling.
(a) Coupling of distribution functions (d.f.) or of probability measures.
For 
- d.f., their coupling is a construction
of two r.v.'s 
and 
on a common probability space.
The same - for more than two r.v.'s.
(b) Coupling of two random variables.
Let 
be defined on 
and 
be defined on 
.
Their coupling: 
and 
on it: 
, 
.
| Lemma 0. | If ( all  and  are d.f. ),
then  a coupling of  and : |
Proof. For a d.f. , define 
:
Put 
(0,1), 
- 
-algebra of
Borel subsets in (0,1), - Lebesgue measure on
(0,1). (?![0,1]?!)
Set 
, 
. Then 
. (?[0,1]?)
Define 
and show . Note: 
In order to avoid some technicalities, assume, for simplicity, that all d.f. are continuous. Put
.
Then 
, 
.
Indeed,
Similarly,
.
Since 
and 
(by definition), then
it is sufficient to show that, for instance, 
.
But both 
and 
are monotone!
And 
a.s., that is 
exists: 
a.s., 
a.s.
If 
, then there exists
point :
But 
!
QDE
| Problem No.1. Prove this lemma without the additional assumption that all d.f. are continuous. |
NEXT SECTION: Uniform Integrability
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