JR
Proof.
Since
X
and
Y
are independent, we know that
C((X, Y)) = \i
x
v,
i.e. the distribution of the ordered pair
(X, Y)
is equal to product measure.
Given
a
Borel subset
H
C
R, let
B =
{(x,y)
G R
2
;
x + y
€
H).
Then
using Fubini's Theorem, we have
P(X + YeH)=
P((X,Y)eB)
=
(/ixi/)(fl)
=
LR^*^)
=
In(lR
1
B{x,y)fJ(dx))i^(dy)
=
JR
viz
G
R;
(x, y)
G
B} u{dy)
=
In
^(
H

y)
v
(
d
y)
so
C(X + Y)=/j,*v.
If
u.
has density
/
and
v
has density
g,
then using
Proposition 6.2.3, shift invariance of Lebesgue measure as in (1.2.5), and
Fubini's theorem again,
(n*v)(H)
=
!^(Hy)
V
(dy)
=
InilHy
1
Kdx))u{dy)
=
J
R
{I
H
_
y
f(x)X(dx))g(y)X(dy)
=
IR(IH
f(
x

v)
K
dx
))a{y) Hd
y
)
=
/R
(
IH
f(
x

y) 9{y)
\(dx))\(dy)
=
I
H
(Iiif(
x
y)9(y)Kdy))x(dx)
=
I
H
(f
*
9)(x) X(dx),
so /i *
v
has density given by
/
*
g.
I
9.5.
EXERCISES.
113
9.5.
Exercises.
Exercise 9.5.1.
For the "simple counterexample" with
tt =
N, P(w) =
2~
w
for w e l l , and
X
n
(ui)
=
2
n
<L,n, verify explicitly that the hypotheses
of each of the Monotone Convergence Theorem, the Bounded Convergence
Theorem, the Dominated Convergence Theorem, and the Uniform Integra
bility Convergence Theorem, are all violated.
Exercise 9.5.2.
Give an example of a sequence of random variables which
is unbounded but still uniformly integrable. For bonus points, make the
sequence also be undominated, i.e. violate the hypothesis of the Dominated
Convergence Theorem.
Exercise 9.5.3.
Let
X,Xi,X%,...
be nonnegative random variables,
defined jointly on some probability triple
(fl,J,
P), each having finite ex
pected value.
Assume that lim
n
_
00
X„(w) =
X(ui)
for all
cje.fl.
For
n,if 6 N, let
Y
n
,K = mm(X
n
,K).
For each of the following statements,
either prove it must true, or provide a counterexample to show it is some
times false.
(a) limi^oo lim
n
^oo E(Y
n?K
) =
B(X).
(b) limn^oo limx^oo E(y
n>K
) = E(X).
Exercise 9.5.4.
Suppose that
lim.
n
^oo X
n
(u>) =
0 for all
w £ fl,
but
lim^oo E[X„] ^ 0. Prove that E(sup
n
X
n
) = oo.
Exercise 9.5.5.
Suppose sup
n
E(X„
r
) < oo for some r > 1. Prove that
{X
n
}
is uniformly integrable. [Hint: If X
n
(cA>) >
a >
0, then X
n
(u;) <
\X
n
{u)\
r
/
a
r
~\]
Exercise 9.5.6.
Prove that Theorem 9.1.6 implies Theorem 9.1.2. [Hint:
Suppose
\X
n
\ < Y
where
E(Y) <
oo. Prove that
{X
n
}
satisfies (9.1.4).]
Exercise 9.5.7.
Prove that Theorem 9.1.2 implies Theorem 4.2.2, assum
ing that EX < oo. [Hint: Suppose
{X
n
}
/
X
where EX < oo. Prove
that
{Xn)
is dominated.]
Exercise 9.5.8.
Let
fl =
{1,2},
with P({1}) = P({2}) =
\,
and let
F
t
({l}) =
t
2
and
F
t
({2}) = t
A
for 0 <
t
< 1.
(a) What does Proposition 9.2.1 conclude in this case?
(b) In light of the above, what rule from calculus is implied by Proposi
tion 9.2.1?
Example 9.5.9.
Let
X
1
,X
2
,.
be
i.i.d.,
each with
P(Xi =
1) =
P(X
t
=
1) = 1/2.
114
9.
MORE PROBABILITY THEOREMS.
(a) Compute the moment generating functions
Mxi^s).
(b) Use Theorem 9.3.4 to obtain an exponentiallydecreasing upper bound
on P ( i ( X i + . . . + * „ ) > 0.1).
Exercise
9.5.10.
Let
Xi,X2,..
be
i.i.d.,
each having the standard
normal distribution
N{0,
1).
Use Theorem 9.3.4 to obtain an exponentially
decreasing upper bound on P
(^(Xi
+
• • •
+
X
n
) >
O.l). [Hint: Don't for
get (9.3.2).]
Example
9.5.11.
Let
X
have the distribution Exponential(5), with
density
fx(x)
=
5
e~
bx
for
x >
0 (with
fx{x)
= 0 for
x <
0).
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