Let me now turn to a foundational result
in the theory of random walks

known as the central limit theorem.

So let's imagine thinking about a random
walk in arbitrary spatial dimensions

in which the single step is given by
a displacement, little x, and the

probability distribution of these single

step displacements is given by little p
of little x.

Let us assume that these little x's are 
iid variables. This is a standard acronym

for independent, identically distributed.

So, independent and identically 
distributed.

So that is, for each x, p of x 
is the same distribution function.

Let us further assume that the mean 
displacement in a single step,

so we no longer have to restrict 
ourselves to a symmetrical random walk.

Let me assume that the mean displacement
in a single step is finite,

very mild restriction.

And let me also assume that the mean 
square displacement in a single step

is also finite.

Under these conditions, then the 
probability distribution after n steps,

so P of n after n steps, of the 
total displacement.

So here, big X is the sum of the 
displacements, little x, n of these guys.

Then, this probability distribution 
of the total displacement is equal to

one over square root of 2 pi n sigma 
squared, e to the minus x, minus n times

the average of little x squared, 
divided by 2, 2 n sigma squared,

where here sigma squared is equal to the 
variance in the single step distribution.

So, it's the difference between x squared
average minus x average squared.

So, this statement is known as the 
central limit theorem.

And I should say that it's holding and the
limit is n is going to infinity.

This is a very powerful and general result
because one needs then only the first two

moments of the single step distribution,
these are the only things that matter to

determine the long distance, long-time
properties of a random walk.

So, this is an example of a universal
statement in which short-range details of

the microscopic process, namely the 
details of the single steps basically do

not matter in the long-time limit.

It doesn't matter if we are talking about
a nearest neighbor random walk in one

dimension or some other type of hopping
distribution as long as the distribution

obeys these two conditions, that the first
and second moments are finite, then the

distribution of the long-time limit is 
this universal Gaussian form.

So, notice also that from this result, 
we now know that the mean displacement

after n steps of a random walk, which is 
just the integral of x p n of x, d x is

equal to n times the displacement 
after a single step.

And similarly, the variance after many
steps of the random walk, x squared

average minus x average squared.

Again using the same definition as before,
so this is the integral of x squared

p n of x, d x minus x average squared, 
which we've already computed.

This thing is simply equal to n sigma 
squared. So, once we know the first and

second moments of the single step 
distribution, what they are, then we know

that the distribution of a random walk of 
n steps is given by a Gaussian form,

and from there we can compute any 
moment we want of the random walk itself.