766 CHAPTER 28. THE NORMAL DISTRIBUTION

and so E(X4)= 3σ4. By now you can see the pattern. If you continue this way, you find

the odd moments are all 0 and

E(X2m)=Cm

(σ

2)m. (28.7)

This is an important observation. In the case of X a random vector, you have φX (t) ≡E (exp(it ·X)) and by taking d

dt j, you can follow the above procedure to obtain E (X j) and

then by using successive differentiations, you can find E (Xni ) or any polynomial in the Xi

assuming the expectations exist.

28.4 Prokhorov and Levy TheoremsRecall one can define the characteristic function of a probability measure µ as

∫Rp eit·xdµ .

In a sense it is more natural. One can also generalize to replace Rp with E a Banach spacein which the dot product t ·x is replaced with t (x) where t ∈ E ′. However, the main interesthere is in Rp.

Definition 28.4.1 A set of Borel probability measures {µn}∞

n=1 defined on a Polishspace E is called “tight” if for all ε > 0 there exists a compact set, Kε such that

µn ([x /∈ Kε ])< ε

for all µn.

How do you determine in general that a set of probability measures is tight?

Lemma 28.4.2 Let E be a separable complete metric space and let Λ be a set of Borelprobability measures. Then Λ is tight if and only if for every ε > 0 and r > 0 there exists afinite collection of balls, {B(ai,r)}m

i=1 such that

µ

(∪m

i=1B(ai,r))> 1− ε

for every µ ∈ Λ.

Proof: If Λ is tight, then there exists a compact set, Kε such that

µ (Kε)> 1− ε

for all µ ∈ Λ. Then consider the open cover, {B(x,r) : x ∈ Kε} . Finitely many of thesecover Kε and this yields the above condition.

Now suppose the above condition and let

Cn ≡ ∪mni=1B(an

i ,1/n)

satisfy µ (Cn) > 1− ε/2n for all µ ∈ Λ. Then let Kε ≡ ∩∞n=1Cn. This set Kε is a compact

set because it is a closed subset of a complete metric space and is therefore complete, andit is also totally bounded by construction. For µ ∈ Λ,

µ(KC

ε

)= µ

(∪nCC

n)≤∑

nµ(CC

n)< ∑

n

ε

2n = ε

Therefore, Λ is tight. ■In case the Polish space is Rp, the case of most interest, there is a very nice condition

interms of characteristic functions which gives “tightness”.