The central limit theorem is one of the most marvelous theorems in mathematics. It can
be proved through the use of characteristic functions. Recall for x ∈ ℝ^{p},
||x ||∞ ≡ max {|xj|,j = 1,⋅⋅⋅,p} .
Also recall the definition of the distribution function for a random vector, X.
FX (x) ≡ P (Xj ≤ xj,j = 1,⋅⋅⋅,p).
Definition 58.18.1Let
{Xn }
be random vectors with values in ℝ^{p}. Then
{λXn}
_{n=1}^{∞}is called “tight”if for all ε > 0 there exists a compact set, K_{ε}suchthat
λXn ([x ∕∈ K ε]) < ε
for all λ_{Xn}. Similarly, if
{μn }
is a sequence of probability measures defined on the Borelsets of ℝ^{p}, then this sequence is “tight” if for each ε > 0 there exists a compact set, K_{ε}such that
μn ([x ∕∈ K ε]) < ε
for all μ_{n}.
Lemma 58.18.2If
{Xn }
is a sequence of random vectors with values in ℝ^{p}suchthat
lnim→∞ ϕXn (t) = ψ(t)
for allt,where ψ
(0)
= 1 and ψ is continuous at0,then
{λXn }
_{n=1}^{∞}is tight.
Proof: Let e_{j} be the j^{th} standard unit basis vector.
||1∫ u ||
||-- (1− ϕXn (tej))dt||
|u∫−uu( ∫ ) |
= ||1- 1 − eitxjdλX dt||
|u − u ℝp n |
| ∫ (∫ ) |
= ||1 u (1 − eitxj)dλ dt||
|u −u ℝp Xn |
||∫ 1∫ u ( ) ||
= || p u- 1− eitxj dtdλXn (x)||
ℝ − u
whenever n is large enough by Lemma 58.18.3 because ψη ∈S. This establishes the
conclusion of the lemma in the case where ψ is also infinitely differentiable. To consider
the general case, let ψ only be uniformly continuous and let ψ_{k} = ψ ∗ ϕ_{k} where ϕ_{k} is a
mollifier whose support is in
(− (1∕k),(1∕k))
^{p}. Then ψ_{k} converges uniformly to ψ and so
the desired conclusion follows for ψ after a routine estimate. This proves the
lemma.
Definition 58.18.5Let μ be a Radon measure on ℝ^{p}. A Borel set, A, is a μcontinuity setif μ
(∂A )
= 0 where ∂A ≡A∖int
(A )
andintdenotes the interior.
The main result is the following continuity theorem. More can be said about the
equivalence of various criteria [?].
Theorem 58.18.6If ϕ_{Xn}
(t)
→ ϕ_{X}
(t)
then λ_{Xn}
(A)
→ λ_{X}
(A)
whenever A is aλ_{X}continuity set.
Proof:First suppose K is a closed set and let
+
ψk (x) ≡ (1 − kdist (x,K )) .
Thus, since K is closed lim_{k→∞}ψ_{k}
(x)
= X_{K}
(x)
. Choose k large enough that
∫
ψkdλX ≤ λX (K)+ ε.
ℝp
Then by Lemma 58.18.4, applied to the bounded uniformly continuous function
ψ_{k},
As an application of this theorem the following is a version of the central limit
theorem in the situation in which the limit distribution is multivariate normal. It
concerns a sequence of random vectors,
{Xk}
_{k=1}^{∞}, which are identically distributed,
have finite mean m, and satisfy
( )
E |Xk|2 < ∞. (58.18.41)
(58.18.41)
Theorem 58.18.7Let
{Xk}
_{k=1}^{∞}be random vectors satisfying 58.18.41, which areindependent and identically distributed with mean m and positive definite covariance
Σ ≡ E
( ∗)
(X − m)(X − m )
. Let
∑n Xj-− m-
Zn ≡ √n- . (58.18.42)
j=1
(58.18.42)
Then forZ ∼N_{p}
(0,Σ )
,
nli→m∞ FZn (x) = FZ (x ) (58.18.43)
(58.18.43)
for allx.
Proof:The characteristic function of Z_{n} is given by
( ) ( ( ))
it⋅∑nj=1 Xj−√mn- ∏n it⋅ Xj√−nm
ϕZn (t) = E e = E e .
j=1
By Taylor’s theorem applied to real and imaginary parts of e^{ix}, it follows
ix x2
e = 1+ ix− f (x) 2
where
|f (x)|
< 2 and
lim f (x) = 1.
x→0
Denoting X_{j} as X, this implies
it⋅(X−√m) X − m ( ( X − m )) (t⋅(X − m))2
e n = 1+ it⋅-√--- − f t⋅ --√--- ------------
n n 2n