Autocorrelation when $m = 0$

Question

I am trying to understand something with respect to autocorrelation. If the process is iid, I can say:

$R_{XX}[n, n + m] = E[x[n]] \times E[x[n + m]] \quad \textrm{if} \quad m \ne 0$

But if $m= 0$

We should get:

$R_{XX}[n, n + m] = E[x[n]] \times E[x[n]] \quad \textrm{if} \quad m = 0$

That is

$R_{XX}[n, n + m] = (E[x[n]])^2$

But in text books I always see, for m = 0

$R_{XX}[n, n + m] = E[x^2[n]]$

Why is it the latter rather than the former?

Answer 1

The general formula is $R_{XX}[n, n + m] = E[x[n] x[n + m]],$ which you can always use.

Specifically, $(x[n],x[n])$ is not an independent pair so you use the general formula.

But for i.i.d. the equation reduces to what you have when $m\neq 0.$