HOMEWORK 2 SOLUTIONS
Problem 1
Fix
i
1
,i
2
,...,i
k
∈ I
.
We shall show that
Y
i
1
,...Y
i
k
are independent by showing that
P
{
Y
i
1
≤
t
i
,...,Y
i
k
≤
t
k
}
=
k
Y
l
=1
P
(
Y
i
l
≤
t
l
)
,
∀
t
1
,t
2
,..t
k
∈
R
.
Step1:
For any bounded continuous functions
f
1
,f
2
,...,f
k
, we have
E
k
Y
l
=1
f
l
(
Y
i
l
)
!
=
k
Y
l
=1
E
f
l
(
Y
i
l
)
.
X
n,i
l
a.s.
→
Y
i
l
implies that
f
l
(
X
n,i
l
)
a.s.
→
f
l
Y
i
l
,
and since all the functions are bounded,
Bounded Convergence Theorem (or DCT) can be applied:
E
k
Y
l
=1
f
l
(
Y
i
l
)
!
= lim
n
→
+
∞
E
k
Y
l
=1
f
l
(
X
n,i
l
)
!
= lim
n
→
+
∞
k
Y
l
=1
E
f
l
(
X
n,i
l
)
!
=
k
Y
l
=1
E
f
l
(
Y
i
l
)
,
where
the equality before last comes from the independence of
{
X
n,i
l
}
k
l
=1
.
Step 2:
Consider the function
f
l,m
=
1
,
if
 ∞ ≤
x
≤
t
l

m
(
x

t
l
) + 1
,
if
t
l
≤
x
≤
t
l
+
m

1
0
,
if
t
l
+
m

1
≤
x <
+
∞
Then
f
l,m
(
x
)
m
→∞
→
1
(
∞
,t
l
]
(
x
)
.
All functions are bounded by 1