Asymptotic Methods in Statistical Inference - Statistics Centre

118
• Multivariate Central Limit Theorem: LetX 1 X :
× 1 be i.i.d. with mean ξ and covariance matrix
Σ. Then √ ³ ¯X − ξ´ → (0 Σ).
Proof: We must show (why?) that
√ ³
t
0 ¯X − t 0 ³
ξ´ → 0 t 0 Σt´
for every t. ThisistheunivariateCLT:put =
t 0 X ; these are i.i.d. with mean t 0 ξ and variance
t 0 Σt and so √ ³ ¯ −t 0 ξ´ ¡
→ 0 t 0 Σt ¢ . But
¯ = t 0 ¯X.
• In the above if Σ 0 then the continuous function
h√
³
¯X − ξ´i 0
Σ
−1 h √
³
¯X − ξ´i → Y 0 Σ −1 Y
where Y ∼ (0 Σ). But Y 0 Σ −1 Y = Z 0 Z for
Z = Σ −12 Y ∼ (0 I). The elements are
i.i.d. (0 1) and so
³ ¯X − ξ´0
Σ
−1 ³ ¯X − ξ´ →
X
=1
2 ∼ 2