Statistics Graduate Level Quiz
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CONVERGENCE & LIMIT THEOREMS
Question 1
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Let
{
A
n
:
n
≥
2
}
\{A_n:n\ge 2\}
{
A
n
:
n
≥
2
}
be independent events satisfying
P
(
A
n
)
=
1
/
n
P(A_n)=1/n
P
(
A
n
)
=
1/
n
, and define
X
n
=
1
A
n
X_n=\mathbf{1}_{A_n}
X
n
=
1
A
n
. Which statement correctly describes the convergence of
X
n
X_n
X
n
?
A
X
n
→
0
X_n\to 0
X
n
→
0
almost surely and therefore also in probability.
B
X
n
→
0
X_n\to 0
X
n
→
0
in probability but not almost surely.
C
X
n
→
1
X_n\to 1
X
n
→
1
in probability but not almost surely.
D
X
n
X_n
X
n
converges neither in probability nor almost surely.
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