Linear Predictive coding vs AR modeling

I'm looking for a suitable explanation of the circumstances in which the LPC error polynomial for a discrete time process x[n] is replaceable with an error polynomial categorized under the AR model? I found this book which states that LPC reduces to AR if the error becomes predominantly white noise(i.e by increasing the number of previous samples in the summation on the RHS of the prediction error equation: $$e[n]=\sum_{i=1}^N a_nx[n-i]+ x[n]$$ Is this the only condition or am I missing something out?

Also, I can't seem to figure out why the complex conjugate of the AR optimal coefficients is taken when moulding an AR process into the LPC form. in the first picture(Th 5.1) note the coefficient derived is $C_n$ and not Cn(complex conjugate). But in Eq 5.5 coefficient used is Cn

I apologize for any technical terminologies I might be messing up and things I haven't attentively read. Being relatively new to this part of the subject, any edits are appreciated. Thank you.

Chapter 5 of this is what my queries pertain to https://authors.library.caltech.edu/25063/1/S00086ED1V01Y200712SPR003.pdf