Mathematical Modeling in Finance with Stochastic Processes

126 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES

Proof.Since the mean of a sum of random variables is the sum of the means,
and scalars factor out of expectations:

E[Sn/n] = (1/n)

∑n

i=1

E[Xi] = (1/n)(nμ) =μ.

Since the variance of a sum ofindependentrandom variables is the sum of
the variances, and scalars factor out of variances as squares:

Var [Sn/n] = (1/n^2 )

∑n

i

Var [Xi] = (1/n^2 )(nσ^2 ) =σ^2 /n.

Fix a value >0. Then using elementary definitions for probability measure
and Chebyshev’s Inequality:

0 ≤Pn[|Sn/n−μ|> ]≤Pn[|Sn/n−μ|≥]≤σ^2 /(n^2 ).

Then by the squeeze theorem for limits

lim

n→∞

P[|Sn/n−μ|> ] = 0.

Jacob Bernoulli originally proved the Weak Law of Large Numbers in 1713
for the special case when theXiare binomial random variables. Bernoulli
had to create an ingenious proof to establish the result, since Chebyshev’s
inequality was not known at the time. The theorem then became known as
Bernoulli’s Theorem. Simeon Poisson proved a generalization of Bernoulli’s
binomial Weak Law and first called it the Law of Large Numbers. In 1929
the Russian mathematician Aleksandr Khinchin proved the general form of
the Weak Law of Large Numbers presented here. Many other versions of the
Weak Law are known, with hypotheses that do not require such stringent
requirements as being identically distributed, and having finite variance.

The Strong Law of Large Numbers

Theorem 6(Strong Law of Large Numbers).LetX 1 ,X 2 ,X 3 ,...,be indepen-
dent, identically distributed random variables each with meanμand variance
E

[

Xj^2

]

<∞. LetSn=X 1 +···+Xn. ThenSn/nconverges with probability
1 toμ,

P

[

lim

n→∞

Sn

n

=μ

]

= 1.