Mathematical Modeling in Finance with Stochastic Processes

146 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES

Key Concepts

1. The probability that the number of heads exceeds the number of tails or
   the number of tails exceeds the number of heads in a sequence of coin-
   flips by some fixed amount can be estimated with the Central Limit
   Theorem and the probability gets close to 1 as the number of tosses
   grows large.


2. The probability that the number of heads exceeds the number of tails
   or the number of tails exceeds the number of heads in a sequence of
   coin-flips by some fixed proportion can be estimated with the Central
   Limit Theorem and the probability gets close to 0 as the number of
   tosses grows large.

Vocabulary

1. Thehalf-integer correction, also called thecontinuity correction
   arises because the distribution of the binomial distribution is a discrete
   distribution, while the standard normal distribution is a continuous
   distribution.

Mathematical Ideas

Introduction

Probability theory generally has two classes of theorems about the results of
coin-tossing games and therefore random walks:

1. Those theorems that tell how well-behaved and natural are the out-
   comes of typical coin-tossing games and random walks. The Weak Law
   of Large Numbers, the Strong Law of Large Numbers and the Central
   Limit Theorem fall into this category.


2. Those theorems that tell how strange and unnatural are the outcomes
   of typical coin-tossing games and random walks. The Arcsine Law and
   the Law of the Iterated Logarithm are good examples in this category.

In this section we will ask two related questions about the net fortune in
a coin-tossing game: