1926 CHAPTER 59. BASIC PROBABILITY

59.19 Characteristic Functions, Prokhorov TheoremRecall one can define the characteristic function of a probability measure. In a sense it ismore natural.

Definition 59.19.1 Let µ be a probability measure defined on the Borel sets of Rp. Then

φ µ (t)≡∫Rp

eit·xdµ.

Also {µn}∞

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

µn ([x /∈ Kε ])< ε

for all µn.

Then there is a version of Lemma 59.18.2 whose proof is identical to the proof of thatlemma.

Lemma 59.19.2 If {µn}is a sequence of Borel probability measures defined on the Borelsets of Rpsuch that

limn→∞

φ µn(t) = ψ (t)

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

n=1 is tight.

Proof: Let e j be the jth standard unit basis vector. Letting t = te j in the definition,∣∣∣∣1u∫ u

−u

(1−φ µn

(te j))

dt∣∣∣∣ (59.19.46)

=

∣∣∣∣1u∫ u

−u

(1−

∫Rp

eitx j dµn (x))

dt∣∣∣∣

=

∣∣∣∣1u∫ u

−u

(∫Rp

(1− eitx j

)dµn (x)

)dt∣∣∣∣

=

∣∣∣∣∫Rp

1u

∫ u

−u

(1− eitx j

)dtdµn (x)

∣∣∣∣=

∣∣∣∣2∫Rp

(1−

sin(ux j)

ux j

)dµn (x)

∣∣∣∣≥ 2

∫[|x j|≥ 2

u ]

(1− 1∣∣ux j

∣∣)

dµn (x)

≥ 2∫[|x j|≥ 2

u ]

(1− 1|u|(2/u)

)dµn (x)