Page 285 - Probability and Statistical Inference

262 5. Concepts of Stochastic Convergence

definition of g’(θ) itself. Recall that for some fixed x, one has:
. We again apply Slutsky’s Theorem, part (ii), in con-
junction with (5.3.6) to complete the proof. !
Remark 5.3.1 Suppose that the X’s are as in the CLT. Then we immedi-
ately conclude that as n → ∞, once we plug in
2
g(x) = x , x ∈ ℜ and apply the Mann-Wald Theorem. We had shown this result
in a different way in the Example 5.3.7. Now, we can also claim that
as n → ∞, by plugging in g(x) = x , x ∈
q
ℜ, for any fixed non-zero real number q (≠ 1). We leave verification of this
result as an exercise. Of course, we tacitly assume that q is chosen in such a
way that both and µ remain well-defined. Note that for negative values of
q
q, we need to assume that the probability of being exactly zero is in fact
zero and that µ is also non-zero.
Example 5.3.8 Let X , ..., X be iid Poisson (λ) with λ > 0. Observe that
1 n
2
in this case µ = σ = λ and then, by the CLT, N(0, 1) as
n → ∞. Then, by using the Remark 5.3.1 and Mann-Wald Theorem, we
immediately see, for example, that as n → ∞. !
Now we move towards an appropriate CLT for the standardized sample
variance. This result again utilizes a nice blend of the CLT (Theorem 5.3.4)
and Slutsky’s Theorem.
Theorem 5.3.6 (Central Limit Theorem for the Sample Variance) Let
2
X , ..., X be iid random variables with mean µ, variance σ (> 0), µ = E{(X 1
4
n
1
4
4
– µ) }, and we assume that 0 < µ < ∞ as well as µ > σ . Denote
4 4
for n ≥ 2. Then,
we have:
Proof Let us first work with . We denote Y = (X –
i i
µ) , i = 1, ..., n, , and write
2

From (5.3.7), observe that