Linear Predictive coding vs AR modeling
LPC reduces to AR modelling only if the stochastic time process is stationary (does not change distribution parameters over time) and ergodic (average over time is equivalent to mean of ensemble average).

This connection between auto-regressive coefficients and the autocovariance of the process is described by the Yule-Walker-Equations (mentioned in the book you linked).

The filter coefficients in your book are defined as conjugates to make definitions more easier (as mentioned below equation 2.5, e.g. equation 2.10).