Mathematical Modeling in Finance with Stochastic Processes

138 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES

Remark.The Central Limit Theorem is true even under the slightly weaker
assumptions thatX 1 ,...Xnonly are independent and identically distributed
with finite meanμand finite varianceσ^2 without the assumption that mo-
ment generating function exists. However, the proof below using moment
generating functions is simple and direct enough to justify using the addi-
tional hypothesis.

Proof.Assume at first thatμ= 0 andσ^2 = 1. Assume also that the moment
generating function of theXi, (which are identically distributed, so there is
only one m.g.f) isφX(t), exists and is everywhere finite. Then the m.g.f of
Xi/

√

nis

φX/√n(t) =E

[

exp(tXi/

√

n)

]

=φX(t/

√

n).

Recall that the m.g.f of a sum ofindependent r.v.s is the product of the
m.g.f.s. Thus the m.g.f ofSn/

√

nis (note that here we used μ= 0 and
σ^2 = 1)
φSn/√n(t) = [φX(t/

√

n)]n

The quadratic approximation (second-degree Taylor polynomial expansion)
ofφX(t) at 0 is by calculus:

φX(t) =φX(0) +φ′X(0)t+ (φ′′X(0)/2)t^2 +r 3 (t) = 1 +t^2 /2 +r 3 (t)

again since E[X] = φ′(0) is assumed to be 0 and Var [X] = E[X^2 ]−
(E[X])^2 = φ′′(0)−(φ′(0))^2 =φ′′(0) is assumed to be 1. Here r 3 (t) is an
error term such that limt→ 0 r 3 (t)/t^2 = 0. Thus,

φ(t/

√

n) = 1 +t^2 /(2n) +r 3 (t/

√

n)

implying that
φSn/√n= [1 +t^2 /(2n) +r 3 (t/

√

n)]n.

Now by some standard results from calculus,

[1 +t^2 /(2n) +r 3 (t/

√

n)]n→exp(t^2 /2)

asn→ ∞. (If the reader needs convincing, it’s computationally easier to
show that
nlog(1 +t^2 /(2n) +r 3 (t/

√

n))→t^2 / 2 ,

using L’Hospital’s Rule in order to account for ther 3 (t) term.)
To handle the general case, consider the standardized random variables
(Xi−μ)/σ, each of which now has mean 0 and variance 1 and apply the
result.