Mathematical Modeling in Finance with Stochastic Processes

4.3 The Central Limit Theorem

Key Concepts

1. The statement, meaning and proof of the Central Limit Theorem.


2. We expect the normal distribution to arise whenever the numerical
   description of a state of a system results from numerous small random
   additive effects, with no single or small group of effects dominant.

Vocabulary

1. TheCentral Limit Theorem: Suppose that for a sequence of inde-
   pendent, identically distributed random variablesXi, eachXihas finite
   varianceσ^2. Let

Zn= (Sn−nμ)/(σ

√

n) = (1/σ)(Sn/n−μ)

√

n

and letZbe the “standard” normally distributed random variable with

mean 0 and variance 1. ThenZnconverges in distribution to Z, that

is:

lim

n→∞

Pr[Zn≤a] =

∫a

−∞

1

√

2 π

exp(−u^2 /2)du

In words, a shifted and rescaled sample distribution is approximately

standard normal.

Mathematical Ideas

Convergence in Distribution

Lemma 12. LetX 1 ,X 2 ,...be a sequence of random variables having cu-
mulative distribution functionsFXn and moment generating functionsφXn.
Let X be a random variable having cumulative distribution function FX
and moment generating function φX. IfφXn(t) → φX(t), for all t, then
FXn(t)→FX(t)for alltat whichFX(t)is continuous.

We say that the sequenceXiconverges in distributiontoXand we
write
Xi→DX.

Notice thatP[a < Xi≤b] =FXi(b)−FXi(a)→F(b)−F(a) =P[a < X≤b],
so convergence in distribution implies convergence of probabilities of events.