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MATH 587
Assignment 4
Due November 30, 2004
(1) Let
X
∈
L
2
(Ω
,
F
,P
) and let
G
be a
σ
subalgebra of
F
. Show that among all
G
measurable r.v.’s
Y
∈
L
2
(Ω
,
F
), there is a unique (a.s.) r.v.
Y
0
which minimizes
k
X
−
Y
k
2
.
(2) (a) Suppose that
Y
1
,Y
2
,...
are nonnegative random variables and that
∑
∞
n
=1
EY
n
<
∞
. Show that
as
n
→∞
,
Y
n
→
0 a.s.
(b) Use (a) to show that if
X
1
,X
2
is a sequence of iid random variables with mean
µ
, variance
σ
2
, and a Fnite fourth moment, then
S
n
/n
→
µ
a.s. as
n
. (Hint: put
Y
n
=(
S
n
−
nµ
)
4
/n
4
,
where
S
n
=
∑
n
i
=1
X
i
.) This is due to Cantelli.
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 '07
 RomanVershynin
 Probability

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