Auto correlation definition
it's funny, but i have an old copy of O&amp;S and i would have expected them to be consistent and use "$x[n]$" notation instead of "$x_n$" notation for discrete-time signals.

The expression above

$$ \phi_{xx}[n,m] \ = \ E[\mathbf{x_n \ x^*_m}] \ = \ \int_\infty^\infty \int_\infty^\infty \ x_n \ x^*_m \ p_{x_n,x_m}(x_n,n,x_m,m) \ dx_n dx_m $$

is the average value of the pair $\mathbf{x_n \ x^*_m}$ where the average is computed from the joint probability density function $ p_{x_n,x_m}(x_n,n,x_m,m) $

If the process is stationary and ergodic (and i think that any process that is fully stationary is also ergodic) i think this can be simplified a little

$$ \phi_{xx}[n-m] \ = \ E[\mathbf{x_n \ x^*_m}] \ = \ \int_\infty^\infty \int_\infty^\infty \ x_n \ x^*_m \ p_{x_n,x_m}(x_n,x_m) \ dx_n dx_m $$

now the root meaning of "ergodic" is that probabilistic averages can be replaced with time averages and vise-versa:

$$ \langle \mathbf{x_n \ x^*_{n+k}} \rangle = \lim_{L \to \infty} \frac{1}{2L+1} \sum\limits_{n=-L}^L \mathbf{x_n \ x^*_{n+k}} $$

the process is ergodic if all averages, including the one expressed above, are the same, whether they are probabilistic or computed from the data

$$ \langle \mathbf{x_n \ x^*_{n+k}} \rangle = \phi_{xx}[k] = E[\mathbf{x_n \ x^*_{n+k}}] $$