Mathematical Modeling in Finance with Stochastic Processes

4.3. THE CENTRAL LIMIT THEOREM 137

wherer 2 (t/n) is a error term such that

lim

n→∞

r(t/n)

(1/n)

= 0.

Then we need to consider

φ

(

t

n

)n

= (1 +μ

t

n

+r 2 (t/n))n.

Taking the logarithm of (1 +μ(t/n) +r(t/n))nand using L’Hospital’s Rule,
we see that
φ(t/n)n→exp(μt).

But this last expression is the moment generating function of the (degenerate)
point mass distribution concentrated atμ. Hence,

lim

n→∞

P[|Sn/n−μ|> ] = 0

The Central Limit Theorem

Theorem 14(Central Limit Theorem).Let random variablesX 1 ,...Xn

• be independent and identically distributed,


• have common meanE[Xi] =μand common varianceVar [Xi] =σ^2 ,


• the common moment generating functionφXi(t) =E[etxi]exists and is
  finite in a neighborhood oft= 0.

ConsiderSn=

∑n

i=1Xi.Let

• Zn= (Sn−nμ)/(σ

√

n) = (1/σ)(Sn/n−μ)

√

n,

• Z be the standard normally distributed random variable with mean 0
  and variance 1.

ThenZnconverges in distribution toZ, that is:

lim

n→∞

P[Zn≤a] =

∫a

−∞

(1/

√

2 π) exp(−u^2 /2)du.