42
CHAPTER 2.
RANDOM VARIABLES
2.3
Independence
Let us first start with the notion of a family of independent random variables.
Definition 2.3.1
[independence of a family of random variables]
Let (Ω
,
F
,
P
) be a probability space and
f
i
: Ω
→
R
,
i
∈
I
, be random
variables where
I
is a non-empty index-set.
The family (
f
i
)
i
∈
I
is called
independent
provided that for all distinct
i
1
, ..., i
n
∈
I
,
n
= 1
,
2
, ...
, and
all
B
1
, ..., B
n
∈ B
(
R
) one has that
P
(
f
i
1
∈
B
1
, ..., f
i
n
∈
B
n
) =
P
(
f
i
1
∈
B
1
)
· · ·
P
(
f
i
n
∈
B
n
)
.
In case, we have a finite index set
I
, that means for example
I
=
{
1
, ..., n
}
,
then the definition above is equivalent to
Definition 2.3.2
[independence of a finite family of random vari-
ables]
Let (Ω
,
F
,
P
) be a probability space and
f
i
: Ω
→
R
,
i
= 1
, . . . , n
,
random variables. The random variables
f
1
, . . . , f
n
are called
independent
provided that for all
B
1
, ..., B
n
∈ B
(
R
) one has that
P
(
f
1
∈
B
1
, ..., f
n
∈
B
n
) =
P
(
f
1
∈
B
1
)
· · ·
P
(
f
n
∈
B
n
)
.
The connection between the independence of random variables and of events
is obvious:
Proposition 2.3.3
Let
(Ω
,
F
,
P
)
be a probability space and
f
i
: Ω
→
R
,
i
∈
I
, be random variables where
I
is a non-empty index-set.
Then the
following assertions are equivalent.
(1)
The family
(
f
i
)
i
∈
I
is independent.
(2)
For all families
(
B
i
)
i
∈
I
of Borel sets
B
i
∈ B
(
R
)
one has that the events
(
{
ω
∈
Ω :
f
i
(
ω
)
∈
B
i
}
)
i
∈
I
are independent.
Sometimes we need to group independent random variables. In this respect
the following proposition turns out to be useful.
For the following we say
that
g
:
R
n
→
R
is
Borel
-measurable (or a Borel function) provided that
g
is (
B
(
R
n
)
,
B
(
R
))-measurable.
Proposition 2.3.4
[Grouping of independent random variables]
Let
f
k
: Ω
→
R
,
k
= 1
,
2
,
3
, ...
be independent random variables.
As-
sume Borel functions
g
i
:
R
n
i
→
R
for
i
= 1
,
2
, ...
and
n
i
∈ {
1
,
2
, ...
}
.
Then the random variables
g
1
(
f
1
(
ω
)
, ..., f
n
1
(
ω
))
,
g
2
(
f
n
1
+1
(
ω
)
, ..., f
n
1
+
n
2
(
ω
))
,
g
3
(
f
n
1
+
n
2
+1
(
ω
)
, ..., f
n
1
+
n
2
+
n
3
(
ω
))
, ... are independent.
The proof is an exercise.