152 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES
vague question precise by asking “How good does the information from the
technical analysis have to be so that the probability of losing money over a
year’s time is 1 in 10,000?”
The 10 stocks over 260 business days in a year means that there 2, 600
daily gains or losses. Denote each daily gain or loss asXi, if the advice is
correct you will gainXi>0 and if the advice is wrong you will loseXi<0.
We want the total changeXannual=
∑ 2600
i=1Xi>0 and we will measure that
by asking thatP[Xannual<0] be small. In the terms of this section, we are
interested in the complementary probability of an excess of successes over
failures.
We assume that the changes are random variables, identically distributed,
independent and the moments of all the random variables are finite. We
will make specific assumptions about the distribution later, for now these
assumptions are sufficient to apply the Central Limit Theorem. Then the
total changeXannual=
∑ 2600
i=1 Xiis approximately normally distributed with
meanμ= 2600·E[X 1 ] and varianceσ^2 = 2600·Var [X 1 ]. Note that here
again we are using the uncentered and unscaled version of the Central Limit
Theorem. In symbols
P
[
a≤(^2600) ∑
i=1
Xi≤b
]
≈
1
√
2 πσ^2∫baexp(
−
(u−μ)^2
2 σ^2)
du.We are interested in
P
[ 2600
∑
i=1Xi≤ 0]
≈
1
√
2 πσ^2∫ 0
−∞exp(
−
(u−μ)^2
2 σ^2)
du.By the change of variablesv= (u−μ)/σ, we can rewrite the probability as
P[Xannual≤0] =1
√
2 π∫−μ/σ−∞exp(
−v^2 / 2)
dv= Φ(
−
μ
σ)
so that the probability depends only on the ratio−μ/σ. We desire that
Φ(−μ/σ) =P[Xannual<0] 1/ 10 ,000. Then we can solve for−μ/σ≈ − 3 .7.
Sinceμ= 2600·E[X 1 ] andσ^2 = 2600·Var [X 1 ], we calculate that for the total
annual change to be a loss we must haveE[X 1 ]≈(3. 7 /
√
2600)·
√
Var [X 1 ] =
0. 07 ·
√
Var [X 1 ].
Now we consider what the requirementE[X 1 ] = 0. 07 ·√
Var [X 1 ] means
for specific distributions. If we assume that the individual changesXiare nor-
mally distributed with a positive mean, then we can useE[X 1 ]/
√
Var [X 1 ] =