Mathematical Modeling in Finance with Stochastic Processes

136 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES

Likewise, convergence of probabilities of events implies convergence in distri-
bution.
This lemma is useful because it is fairly routine to determine the pointwise
limit of a sequence of functions using ideas from calculus. It is usually much
easier to check the pointwise convergence of the moment generating func-
tions than it is to check the convergence in distribution of the corresponding
sequence of random variables.
We won’t prove this lemma, since it would take us too far afield into
the theory of moment generating functions and corresponding distribution
theorems. However, the proof is a fairly routine application of ideas from the
mathematical theory of real analysis.

Application: Weak Law of Large Numbers.

Here’s a simple representative example of using the convergence of the mo-
ment generating function to prove a useful result. We will prove a version of
the Weak Law of Large numbers that does not require the finite variance of
the sequence of independent, identically distributed random variables.

Theorem 13 (Weak Law of Large Numbers).LetX 1 ,...,Xn be indepen-
dent, identically distributed random variables each with meanμand such that
E[|X|]is finite. LetSn=X 1 +···+Xn. ThenSn/nconverges in probability
toμ. That is:

lim

n→∞

P[|Sn/n−μ|> ] = 0

Proof.If we denote the moment generating function ofXbyφ(t), then the
moment generating function of

Sn

n

=

∑n

i=1

Xi

n

is (φ(t/n))n. The existence of the first moment assures us thatφ(t) is dif-
ferentiable at 0 with a derivative equal toμ. Therefore, by tangent-line
approximation (first-degree Taylor polynomial approximation)

φ

(

t

n

)

= 1 +μ

t

n

+r 2 (t/n)