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ACM/EE 116  Fall 2009  Homework 3 Solution
•
Handed out: Nov 3, 2009, Due: Nov 17, 2009 (in class).
•
Please write down your solutions clearly and concisely, box your an
swers, put problems in order and bind pages.
•
Grader: Molei Tao
1. A function
φ
(
x
) is said to belong to the ‘upper class’ if
P
(
S
n
>
φ
(
n
)
√
n
i.o.) = 0. A consequence of the law of the iterated logarithm is
that
√
α
log log
x
is in the upper class for all
α >
2. Use the ﬁrst Borel
Cantelli lemma to prove the much weaker fact that
φ
(
x
) =
√
α
log
x
is in the upper class for all
α >
2, in the special case when the
X
i
are
independent
N
(0
,
1) variables. (Hint: recall that
S
n
=
∑
n
i
=1
X
i
. Only
need to consider
φ
(
x
) evaluated at integer
x
.)
2. Let
Y
be uniformly distributed on [

1
,
1] and let
X
=
Y
2
.
(a) Find the best predictor of X given Y, and of Y given X.
(b) Find the best linear predictor of X given Y, and of Y given X.
(Hint: a best predictor of
A
given
B
is a function
f
(
B
) that minimizes
E
(
A

f
(
B
))
2
. Diﬀerent function spaces to which
f
(
·
) is restricted may
result in diﬀerent best predictors.)
3. (a) Suppose that
X
1
,X
2
,...
is a sequence of random variables, each
having a normal distribution, and such that
X
n
D
→
X
. Show that
X
has a normal distribution, possibly degenerate.
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