Chapter 2: FAQs 35
still works. You are even allowed to have some weak
dependence between the variables.
A generalization that is important in finance applies to
distributions with infinite variance. If the tails of the
individual distributions have a power-law decay,|x|−^1 −α
with 0<α<2 then the average will tend to a stable
Levy distribution. ́
If you add random numbers and get normal, what hap-
pens when you multiply them? To answer this question
we must think in terms of logarithms of the random
numbers.
Logarithms of random numbers are themselves random
(let’s stay with logarithms of strictly positive numbers).
So if you add up lots of logarithms of random numbers
you will get a normal distribution. But, of course, a
sum of logarithms is just the logarithm of a product,
therefore the logarithm of the product must be normal,
and this is the definition of lognormal: the product of
positive random numbers converges to lognormal.
This is important in finance because a stock price after
a long period can be thought of as its value on some
starting day multiplied by lots of random numbers, each
representing a random return. So whatever the distribu-
tion of returns is, the logarithm of the stock price will
be normally distributed. We tend to assume that equity
returns are normally distributed, and equivalently, equi-
ties themselves are lognormally distributed.
References
Feller, W 1968An Introduction to Probability Theory and Its
Applications. 3rd ed. New York, Wiley
