Mathematical Modeling in Finance with Stochastic Processes

4 Limit Theorems for Stochastic Processes

Vocabulary

1. TheWeak Law of Large Numbersis a precise mathematical state-
   ment of what is usually loosely referred to as the “law of averages”.
   Precisely, letX 1 ,...,Xn be independent, identically distributed ran-
   dom variables each with meanμand varianceσ^2. LetSn=X 1 +···+Xn
   and consider thesample meanor more loosely, the “average”Sn/n.
   Then the Weak Law of Large Numbers says that the sample meanSn/n
   converges in probability to the population meanμ. That is:

lim

n→∞

Pn[|Sn/n−μ|> ] = 0

In words, the proportion of those samples whose sample mean differs

significantly from the population mean diminishes to zero as the sample

size increases.

2. TheStrong Law of Large Numberssays thatSn/nconverges toμ
   with probability 1. That is:

P

[

lim

n→∞

Sn/n=μ

]

= 1

In words, the Strong Law of Large Numbers “almost every” sample

mean approaches the population mean as the sample size increases.

Mathematical Ideas

4.1 Laws of Large Numbers

Lemma 3(Markov’s Inequality).IfXis a random variable that takes only

nonnegative values, then for anya > 0 :

P[X≥a]≤E[X]/a

Proof.Here is a proof for the case whereXis a continuous random variable