Mathematical Modeling in Finance with Stochastic Processes

4.1. LAWS OF LARGE NUMBERS 125

with probability densityf:

E[X] =

∫∞

0

xf(x)dx

=

∫a

0

xf(x)dx+

∫∞

a

xf(x)dx

≥

∫∞

a

xf(x)dx

≥

∫∞

a

af(x)dx

=a

∫∞

a

f(x)dx

=aP[X≥a].

(The proof for the case whereX is a purely discrete random variable is
similar with summations replacing integrals. The proof for the general case
is exactly as given withdF(x) replacingf(x)dxand interpreting the integrals
as Riemann-Stieltjes integrals.)

Lemma 4(Chebyshev’s Inequality).IfXis a random variable with finite
meanμand varianceσ^2 , then for any valuek > 0 :

P[|X−μ|≥k]≤σ^2 /k^2.

Proof. Since (X−μ)^2 is a nonnegative random variable, we can apply Markov’s
inequality (witha=k^2 ) to obtain

P

[

(X−μ)^2 ≥k^2

]

≤E

[

(X−μ)^2

]

/k^2.

But since (X−μ)^2 ≥k^2 if and only if|X−μ| ≥k, the inequality above is
equivalent to:
P[|X−μ|≥k]≤σ^2 /k^2

and the proof is complete.

Theorem 5(Weak Law of Large Numbers).LetX 1 ,X 2 ,X 3 ,...,be indepen-
dent, identically distributed random variables each with meanμand variance
σ^2. LetSn=X 1 +···+Xn. ThenSn/nconverges in probability toμ. That
is:
lim
n→∞
P[|Sn/n−μ|> ] = 0.