Statistics Graduate Level Quiz
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CONVERGENCE & LIMIT THEOREMS
Question 1
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Suppose
n
X
n
→
p
λ
nX_n\xrightarrow{p}\lambda
n
X
n
p
λ
for a finite constant
λ
\lambda
λ
, and assume
P
(
X
n
>
−
1
)
→
1
P(X_n>-1)\to1
P
(
X
n
>
−
1
)
→
1
. What is the probability limit of
(
1
+
X
n
)
n
(1+X_n)^n
(
1
+
X
n
)
n
?
A
1
+
λ
,
1+\lambda,
1
+
λ
,
obtained by replacing the power expression with its first-order binomial approximation.
B
e
λ
,
e^\lambda,
e
λ
,
obtained by applying Slutsky's theorem to the logarithm of the expression.
C
e
λ
2
/
2
,
e^{\lambda^2/2},
e
λ
2
/2
,
obtained by retaining the leading quadratic term in the logarithmic expansion.
D
No limit is implied because the transformation
x
↦
(
1
+
x
)
n
x\mapsto(1+x)^n
x
↦
(
1
+
x
)
n
changes with
n
n
n
.
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