59.18. THE CENTRAL LIMIT THEOREM 1919

59.18 The Central Limit TheoremThe central limit theorem is one of the most marvelous theorems in mathematics. It can beproved through the use of characteristic functions. Recall for x ∈ Rp,

||x||∞≡max

{∣∣x j∣∣ , j = 1, · · · , p

}.

Also recall the definition of the distribution function for a random vector, X.

FX (x)≡ P(X j ≤ x j, j = 1, · · · , p) .

Definition 59.18.1 Let {Xn} be random vectors with values in Rp. Then {λ Xn}∞

n=1 iscalled “tight” if for all ε > 0 there exists a compact set, Kε such that

λ Xn ([x /∈ Kε ])< ε

for all λ Xn . Similarly, if {µn} is a sequence of probability measures defined on the Borelsets of Rp, then this sequence is “tight” if for each ε > 0 there exists a compact set, Kε

such thatµn ([x /∈ Kε ])< ε

for all µn.

Lemma 59.18.2 If {Xn}is a sequence of random vectors with values in Rpsuch that

limn→∞

φ Xn(t) = ψ (t)

for all t, where ψ (0) = 1 and ψ is continuous at 0, then {λ Xn}∞

n=1 is tight.

Proof: Let e j be the jth standard unit basis vector.∣∣∣∣1u∫ u

−u

(1−φ Xn

(te j))

dt∣∣∣∣

=

∣∣∣∣1u∫ u

−u

(1−

∫Rp

eitx j dλ Xn

)dt∣∣∣∣

=

∣∣∣∣1u∫ u

−u

(∫Rp

(1− eitx j

)dλ Xn

)dt∣∣∣∣

=

∣∣∣∣∫Rp

1u

∫ u

−u

(1− eitx j

)dtdλ Xn (x)

∣∣∣∣=

∣∣∣∣2∫Rp

(1−

sin(ux j)

ux j

)dλ Xn (x)

∣∣∣∣≥ 2

∫[|x j|≥ 2

u ]

(1− 1∣∣ux j

∣∣)

dλ Xn (x)