Asymptotic Methods in Statistical Inference - Statistics Centre

38
• Lemma: { } =1 independent, with zero mean,
variance 2 , common h¯¯¯ 3¯¯¯i
. Consider a linear
combination = P with P 2 = 1.
Then Liapounov’s Theorem applies, and yields
→ Φ, if
X
| | 3 → 0 (4.2)
=1
Equivalently,
= max | | → 0 (4.3)
1≤≤
Proof: (This is essentially Theorems 2.7.3, 2.7.4
of the text.) In the notation above = ,
2 =
X
X
2 = 2 2 = 2
=1 =1
and so → Φ as long as
1
X
∙¯¯¯¯¯¯3¸
3 = h | | 3i X
=1
3 | | 3 → 0
=1
To see that (4.2) and (4.3) are equivalent, first