PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Hence we have shown M Zn (t) = (1 + a n ) n , where
lim na n = t 2 /2, so by Lemma 1.11.1,
n→∞
lim M Z n
(t) = e t2 /2 for each fixed t.
n→∞
Remarks
1. The Central Limit Theorem can be stated equivalently for ¯X n ,
( ) ¯Xn − µ
i.e., L
σ/ √ → N(0, 1)
n
(
just note that ¯X n − µ
σ/ √ n = S n − nµ
σ √ n
2. The Central Limit Theorem holds under conditions more general than those given
above. In particular, with suitable assumptions,
(i) M X (t) need not exist.
(ii) X 1 , X 2 , . . . need not be i.i.d..
3. Theorems 1.11.1 and 1.11.2 are concerned with the asymptotic behaviour of ¯X n .
)
.
Theorem 1.11.1 states ¯X n → µ in prob as n → ∞.
Theorem 1.11.2 states ¯X n − µ
σ/ √ n −→ D
N(0, 1) as n → ∞.
These results are not contradictory because Var( ¯X n ) → 0, but the Central Limit
Theorem is concerned with ¯X n − E( ¯X n )
√
Var( ¯Xn ) . 76