PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Remarks
This is a simple generalisation of the standard limit,
(
1 +
n) x n
= e x .
lim
n→∞
Consider a sequence of i.i.d. RVs X 1 , X 2 , . . . with E(X i ) = µ, Var(X i ) = σ 2 and such
that the MGF, M X (t), is defined for all t in some open interval containing 0.
n∑
Let S n = X i and note that E[S n ] = nµ and Var(S n ) = nσ 2 .
i=1
Theorem. 1.11.2 (Central Limit Theorem)
Let X 1 , X 2 , . . . , S n be as above and let Z n = S n − nµ
σ √ n
. Then
Proof.
L[Z n ] → N(0, 1) as n → ∞.
We will use the fact that it is sufficient to prove that
M Zn (t) → e t2 /2
for each fixed t
[Note: if Z ∼ N(0, 1) then M Z (t) = e t2 /2 .]
Now let U i = X i − µ
σ
and observe that Z n = 1 √ n
Since the U i are independent we have:
n∑
U i .
i=1
M Zn (t) = { M U (t/ √ n) } n
⎡
⎣
M aX (t) = E[e taX ]
= M X (at)
⎤
⎦
Now,
E(U i ) = 0 ⇒ M ′ U(0) = 0,
Var(U i ) = 1 ⇒ M ′′
U(0) = 1.
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