EE 464, Mitra
Spring 2009
Homework 12
Due FRIDAY, April 24, 2009, 10am
This problem set investigates random sequences: convergence of random sequences and limit theorems.
1.
LG
7.21.
2.
LG
7.45 (Hint: use the CauchySchwarz inequality).
3. Let
ω
∼
U
[0
,
1]
. In what senses do the following sequences converge, it at all? For each sequence,
identify the limit and show whether they converge
in distribution, in probability, in meansquare,
almost surely
or
surely
.
(a)
X
n
= sin
(
ω
+
1
n
)
(b)
X
n
= cos
n
(
ω
)
4. We have a sequence of independent random variables, whose probability density functions for each
n
are given by
f
X
n
(
x
)
=
1

1
n
¶
1
√
2
πσ
exp
"

1
2
σ
2
x

n

1
n
σ
¶
2
#
+
1
n
σ
exp(

σx
)
u
(
x
)
where
u
(
x
)
is the unit stepfunction.
Determine whether or not the sequence converges in (i)
meansquare (ii) probability or (iii) distribution.
5. Consider an
i.i.d.
sequence of uniform random variables:
X
n
∼
U
[0
,
1]
.
Define the random
sequence
Y
n
= min
1
≤
i
≤
n
X
i
.
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 Spring '06
 Caire
 Variance, Gate, Probability theory, probability density function, uniform random variables, WLLN

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