Mathematical Modeling in Finance with Stochastic Processes

150 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES

asn→∞.
RewritingP

[∣∣Tn

n

∣

∣> ]=P[|Tn|> n] this restatement of the Weak Law

actually provides the answer to our second question: The probability that
the absolute excess of heads over tails is greater than a fixed fraction of the
flips in a fair game of durationnapproaches 0 asnincreases.
Finally, this gives an estimate on the central probability in a binomial
distribution.

Corollary 6.

P[Tn= 0]≈P

[

|Z|<(1/2)/

√

n

]

→ 0

asn→∞.

We can estimate this further

P

[

|Z|<(1/2)/

√

n

]

=

1

√

2 π

∫ 1 /(2√n)

− 1 /(2√n)

e−u

(^2) / 2
du

=

1

√

2 π

∫ 1 /(2√n)

− 1 /(2√n)

1 −u^2 /2 +u^4 /8 +... du

=

1

√

2 π

1

√

n

−

1

24

√

2 π

1

n^3 /^2

+

1

640

√

2 π

1

n^5 /^2

+....

So we see thatP[Tn= 0] goes to zero at the rate of 1/

√

n.

Illustration 1

What is the probability that the number of heads exceeds the number of tails
by more than 20 or the number of tails exceeds the number of heads by more
than 20 after 500 tosses of a fair coin? By the proposition, this is:

P[|Tn|>20]≈P

[

|Z|≥ 20. 5 /

√

500

]

= 0. 3477.

This is a reasonably large probability, and is larger than many people would
expect.

Here is a graph of the probability of at leastsexcess heads in 500 tosses
of a fair coin: