3.8.
MODES OF CONVERGENCE
73
Example 3.8.4
Assume ([0
,
1]
,
B
([0
,
1])
, λ
) where
λ
is the Lebesgue mea
sure. We take
f
1
= 1I
[0
,
1
2
)
,
f
2
= 1I
[
1
2
,
1]
,
f
3
= 1I
[0
,
1
4
)
,
f
4
= 1I
[
1
4
,
1
2
]
,
f
5
= 1I
[
1
2
,
3
4
)
,
f
6
= 1I
[
3
4
,
1]
,
f
7
= 1I
[0
,
1
8
)
, . . .
This implies lim
n
→∞
f
n
(
x
)
→
0 for all
x
∈
[0
,
1]
.
But it holds convergence in
probability
f
n
λ
→
0: choosing 0
< ε <
1 we get
λ
(
{
x
∈
[0
,
1] :

f
n
(
x
)

> ε
}
)
=
λ
(
{
x
∈
[0
,
1] :
f
n
(
x
) = 0
}
)
=
1
2
if
n
= 1
,
2
1
4
if
n
= 3
,
4
, . . . ,
6
1
8
if
n
= 7
, . . .
.
.
.
As a preview on the next probability course we give some examples for the
above concepts of convergence. We start with the
weak law of large numbers
as an example of the convergence in probability:
Proposition 3.8.5
[Weak law of large numbers]
Let
(
f
n
)
∞
n
=1
be a
sequence of independent random variables with
E
f
k
=
m
and
E
(
f
k

m
)
2
=
σ
2
for all
k
= 1
,
2
,
. . . .
Then
f
1
+
· · ·
+
f
n
n
P
→
m
as
n
→ ∞
,
that means, for each
ε >
0
,
lim
n
P
ω
:

f
1
+
· · ·
+
f
n
n

m

> ε
→
0
.
Proof
. By Chebyshev’s inequality (Corollary 3.6.9) we have that
P
ω
:
f
1
+
· · ·
+
f
n

nm
n
> ε
≤
E

f
1
+
· · ·
+
f
n

nm

2
n
2
ε
2
=
E
(
∑
n
k
=1
(
f
k

m
))
2
n
2
ε
2
=
nσ
2
n
2
ε
2
→
0
as
n
→ ∞
.
Using a stronger condition, we get easily more: the almost sure convergence
instead of the convergence in probability. This gives a form of the
strong law
of large numbers
.