Tutorials in Probability.pdf

(2725 KB) Pobierz
136899556 UNPDF
Tutorial 1: Dynkin systems
1
1. Dynkin systems
Denition 1 A dynkin system on a set is a subset
D
of the
power set
() , with the following properties:
( i )
2D
( ii ) A;B
2D
;A
B
)
B
n
A
2D
( iii ) A n 2D
;A n
A n +1 ;n
1
)
+ [
A n 2D
n =1
Denition 2 A -algebra on a set is a subset
F
of the power
set
P
() with the following properties:
( i )
2F
( ii ) A
2F )
A c 4 =
n
A
2F
( iii ) A n 2F
;n
1
)
+ [
A n 2F
n =1
P
Tutorial 1: Dynkin systems
2
Exercise 1.
Let
F
be a -algebra on . Show that
;2F
,that
if A;B
2F
then A
[
B
2F
and also A
\
B
2F
. Recall that
B
n
A = B
\
A c and conclude that
F
is also a dynkin system on .
Exercise 2.
Let (
D i ) i2I be an arbitrary family of dynkin systems
=
on , with I
=
;
. Show that
D
\ i2I D i is also a dynkin system on
.
Exercise 3.
Let (
F i ) i2I be an arbitrary family of -algebras on ,
=
with I
=
;
. Show that
F
\ i2I F i is also a -algebra on .
Exercise 4.
Let
A
be a subset of the power set
P
(). Dene:
D (
A
) =
fD
dynkin system on :
ADg
Show that
P
() is a dynkin system on , and that D (
A
)isnotempty.
Dene:
) = \
D
(
A
D2D ( A ) D
Tutorial 1: Dynkin systems
3
), and
that it is the smallest dynkin system on with such property, (i.e. if
D
D
(
A
) is a dynkin system on such that
AD
(
A
is a dynkin system on with
AD
,then
D
(
A
)
D
).
Denition 3 Let
AP
() .Wecall dynkin system generated
by
A
, the dynkin system on ,denoted
D
A
) , equal to the intersection
of all dynkin systems on , which contain
A
.
Exercise 5.
Do exactly as before, but replacing dynkin systems by
-algebras.
Denition 4 Let
AP
() .Wecal -algebra generated by
A
,the-algebra on ,denoted (
A
) , equal to the intersection of all
-algebras on , which contain
A
.
() is cal led a -system
on , if and only if it is closed under nite intersection, i.e. if it has
the property:
A
of the power set
P
A;B
2A )
A
\
B
2A
Show that
(
Denition 5 Asubset
Tutorial 1: Dynkin systems
4
Exercise 6.
Let
A
be a -system on . For all A
2D
(
A
), we dene:
( A ) =
f
B
2D
(
A
): A
\
B
2D
(
A
)
g
1. If A
2A
, show that
A
( A )
2. Show that for all A
2D
(
A
), ( A ) is a dynkin system on .
3. Show that if A
2A
,then
D
(
A
)
( A ).
4. Show that if B
2D
(
A
), then
A
( B ).
5. Show that for all B
2D
(
A
),
D
(
A
)
( B ).
6. Conclude that
D
(
A
)isalsoa -system on .
Exercise 7.
Let
D
be a dynkin system on which is also a -system.
1. Show that if A;B
2D
then A
[
B
2D
.
Tutorial 1: Dynkin systems
5
2. Let A n 2D
;n
1. Consider B n =
[
i =1 A i . Show that
[
1
n =1 A n =
[
1
n =1 B n .
3. Show that
D
is a -algebra on .
Exercise 8.
Let
A
be a -system on . Explain why
D
(
A
)isa
-algebra on , and (
A
) is a dynkin system on . Conclude that
D
(
A
)= (
A
). Prove the theorem:
be a collection of subsets of
which is closed under pairwise intersection. If
C
D
is a dynkin system
containing
C
then
D
also contains the -algebra (
C
) generated by
C
.
+
+
Theorem 1 (dynkin system) Let

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin